What a balance of division by 45. division of integers with the residue, rules, examples
Signs of divisibility numbers- These are rules that allow non-producing divisions relatively quickly find out whether this number is divided into a given without residue.
Some of signs of divisibility Pretty simple, some harder. On this page you will find as signs of divisibility simple numbers, such as, for example, 2, 3, 5, 7, 11, and signs of the divisibility of components, such as 6 or 12.
I hope this information will be useful to you.
Pleasant learning!
Sign of divisibility on 2
This is one of the easiest signs of divisibility. It sounds like this: if the recording of a natural number ends with a reader, then it is evenly (divided without a residue by 2), and if the record of the number ends in an odd digit, then this number is odd.
In other words, if the last digit number is equal 2
, 4
, 6
, 8
or 0
- the number is divided into 2, if not, it is not divided
For example, numbers: 23 4
, 8270
, 1276
, 9038
, 502
They are divided into 2, because they are even.
Numbers: 23 5
, 137
, 2303
On 2 are not divided, because they are odd.
Sign of divisibility on 3
This feature of the division is completely different: if the number of numbers is divided by 3, then the number is divided by 3; If the amount of numbers number is not divided by 3, then the number is not divided by 3.
So, to understand whether the number is divided into 3, it is only necessary to add the numbers among themselves from which it consists.
It looks like this: 3987 and 141 are divided by 3, because in the first case 3 + 9 + 8 + 7 \u003d 27
(27: 3 \u003d 9 - it is divided without the remains of 3), and in the second 1 + 4 + 1 \u003d 6
(6: 3 \u003d 2 - Also divided without the remains of 3).
But the numbers: 235 and 566 are not divided into 3, because 2 + 3 + 5 \u003d 10
and 5 + 6 + 6 \u003d 17
(And we know that neither 10 nor 17 are divided into 3 without a residue).
Sign of divisibility on 4
This sign of divisibility will be more complicated. If the last 2 digits of numbers form the number divided by 4 or it is 00, then the number is divided into 4, otherwise this number is not divided into 4 without a residue.
For example: 1. 00
and 3. 64
divided by 4, because in the first case the number ends on 00
, and in the second 64
which in turn is divided into 4 without a residue (64: 4 \u003d 16)
Numbers 3. 57
and 8. 86
do not divide on 4 because neither 57
n. 86
4 are not divided, and therefore do not correspond to this sign of divisibility.
Sign of divisibility on 5
And again, we have a rather simple sign of divisibility: if the recording of the natural number ends with a number 0 or 5, then this number is divided without a residue by 5. If the number of the number ends with a different digit, then the number without a residue is not divided into 5.
This means that any numbers ending in numbers 0
and 5
, for example, 1235. 5
and 43. 0
, fall a rule and divided by 5.
A, for example, 1549 3
and 56. 4
Do not end on the figure 5 or 0, which means they can not share for 5 without a residue.
Sign of divisibility on 6
We have a composite number 6, which is a product of numbers 2 and 3. Therefore, a sign of divisibility by 6 is also composite: so that the number is divided by 6, it must correspond to two signs of divisibility simultaneously: a sign of divisibility on 2 and a sign of divisibility by 3. At the same time, note that such a composite number as 4 has an individual sign of divisibility, because it is the evidence of the number 2 on itself. But back to the sign of divisibility on 6.
The numbers 138 and 474 are even corresponding to the signs of divisibility by 3 (1 + 3 + 8 \u003d 12, 12: 3 \u003d 4 and 4 + 7 + 4 \u003d 15, 15: 3 \u003d 5), which means they are divided by 6. But 123 and 447, although they are divided into 3 (1 + 2 + 3 \u003d 6, 6: 3 \u003d 2 and 4 + 4 + 7 \u003d 15, 15: 3 \u003d 5), but they are odd, and therefore do not correspond to the sign of divisibility by 2, And therefore, they do not correspond to the sign of divisibility by 6.
Sign of divisibility on 7
This sign of divisibility is more complicated: the number is divided into 7 if the result of subtraction of the twin-lasting figure of tens of this number is divided into 7 or equal to 0.
Sounds quite confusing, but in practice it's easy. See ourselves: number 95
9 is divided into 7, because 95
-2 * 9 \u003d 95-18 \u003d 77, 77: 7 \u003d 11 (77 divided by 7 without residue). And if the number with the number obtained during the transformations arose (due to its size it is difficult to understand, it is divided into 7 or not, then this procedure can be continued as many times as you feel necessary).
For example, 45
5 I. 4580
1 possess signs of divisibility to 7. In the first case, everything is quite simple: 45
-2 * 5 \u003d 45-10 \u003d 35, 35: 7 \u003d 5. In the second case we will do this: 4580
-2 * 1 \u003d 4580-2 \u003d 4578. It is difficult for us to understand whether it is divided if 457
8 to 7, so we repeat the process: 457
-2 * 8 \u003d 457-16 \u003d 441. And again we use a sign of divisibility, because we are still three-digit number 44
1. So 44
-2 * 1 \u003d 44-2 \u003d 42, 42: 7 \u003d 6, i.e. 42 is divided by 7 without a balance, which means it is 45801 divided by 7.
But the numbers 11
1 I. 34
5 are not divided into 7, because 11
-2 * 1 \u003d 11-2 \u003d 9 (9 is not divided without a residue by 7) and 34
-2 * 5 \u003d 34-10 \u003d 24 (24 is not divided without residue by 7).
Sign of divisibility on 8
The sign of divisibility on 8 sounds like this: if the last 3 digits form a number divided by 8, or it is 000, then the specified number is divided by 8.
Numbers 1. 000
or 1. 088
divided by 8: the first ends on 000
, second 88
: 8 \u003d 11 (divided by 8 without a residue).
But number 1 100
or 4. 757
do not divide on 8, since numbers 100
and 757
Do not share without residue.
Sign of divisibility on 9
This sign of divisibility is similar to a sign of divisibility by 3: if the number of numbers is divided by 9, then the number is divided into 9; If the number of numbers is not divided into 9, then the number is not divided by 9.
For example: 3987 and 144 are divided into 9, because in the first case 3 + 9 + 8 + 7 \u003d 27
(27: 9 \u003d 3 - it is divided without the remains of 9), and in the second 1 + 4 + 4 \u003d 9
(9: 9 \u003d 1 - Also divided without the remains of 9).
But the numbers: 235 and 141 are not divided into 9, because 2 + 3 + 5 \u003d 10
and 1 + 4 + 1 \u003d 6
(And we know that neither 10 nor 6 are divided into 9 without a residue).
Signs of divisibility on 10, 100, 1000 and other bit units
These signs of divisibility I combined because they can be described equally: the number is divided into a discharge unit if the number of zeros at the end of the number is greater than or equal to the number of zeros in a given bit one.
In other words, for example, we have such numbers: 654 0
, 46400
, 867000
, 6450
. Of these, everyone is divided into 1 0
; 46400
and 867. 000
They are divided into 1 00
; And only one of them - 867 000
divided by 1. 000
.
Any numbers in which the number of zeros at the end is less than that of the discharge unit, are not divided into this discharge unit, for example 600 30
and 7. 93
Do not share 1. 00
.
Sign of divisibility on 11
In order to find out whether the number is divided into 11, it is necessary to obtain the difference in the sums of even and odd numbers of this number. If this difference is equal to 0 or divided by 11 without a residue, then the number itself is divided by 11 without a residue.
To make it clearer, I propose to consider the examples: 2
35
4 is divided by 11, because ( 2
+5
)-(3+4)=7-7=0. 29
19
4 is also divided into 11, since ( 9
+9
)-(2+1+4)=18-7=11.
But 1. 1
1 or 4
35
4 are not divided by 11, since in the first case we have (1 + 1) - 1
\u003d 1, and in the second ( 4
+5
)-(3+4)=9-7=2.
Sign of divisibility on 12
The number 12 is composite. Its sign of divisibility is the correspondence of the signs of divisibility by 3 and on 4 at the same time.
For example, 300 and 636 correspond to the signs of divisibility on 4 (the last 2 digits are zeros or are divided into 4) and signs of divisibility by 3 (the sum of the numbers and the first and thorough number is divided into 3), and will be applied, they are divided by 12 without a balance.
But 200 or 630 are not divided into 12, because in the first case the number only responds with a sign of divisibility by 4, and in the second - only a sign of divisibility by 3. But not both of the signs at the same time.
Sign of divisibility on 13
The sign of divisibility on 13 is that if the number of tens of numbers, folded with multiplied by 4 units of this number, will be multiple 13 or equal to 0, then the number itself is divided by 13.
Take for example 70
2. So 70
+ 4 * 2 \u003d 78, 78: 13 \u003d 6 (78 is divided without a residue by 13), it means 70
2 is divided by 13 without a residue. Another example is the number 114
4. 114
+ 4 * 4 \u003d 130, 130: 13 \u003d 10. The number 130 is divided into 13 without a residue, which means a given number corresponds to a sign of divisibility by 13.
If you take numbers 12
5 or 21
2, then we get 12
+ 4 * 5 \u003d 32 and 21
+ 4 * 2 \u003d 29 corresponded, and neither 32 nor 29 are divided into 13 without a residue, which means that the specified numbers are not divided without a residue by 13.
Dividitude of numbers
As can be seen from the above, it can be assumed that to any of natural numbers You can choose your individual sign of divisibility or "composite" feature if the number is multiple of several different numbers. But as practice shows, mainly the greater the number, the more difficult it is its sign. Perhaps the time spent on checking a sign of divisibility may be equal to or more than the division itself. Therefore, we usually use the simplest of the signs of divisibility.
In this article we will analyze division of integers with the residue. Let's start with the general principle of dividing integers with the residue, we formulate and prove the theorem about the divisibility of integers with the residue, trace the connection between the divisible, divider, incomplete private and the residue. Then let's voice the rules on which the division of integers with the residue is carried out, and consider the use of these rules when solving examples. After that, learn how to check the result of dividing integers with the residue.
Navigating page.
General view of division of integers with the residue
The division of integers with the residue we will consider as a generalization of division with the residue of natural numbers. This is due to the fact that natural numbers are part of integers.
Let's start with terms and designations that are used in the description.
By analogy with the division of natural numbers with the residue, we will assume that the result of dividing with the residue of two integers A and B (B is not zero) are two integers C and D. Numbers a and b are called divisible and divider Accordingly, the number D - residue from division A on b, and an integer C is called incomplete private (or simply privateif the residue is zero).
We agree to assume that the residue is a non-negative number, and its value does not exceed B, that is, we met, when we were told about the comparison of three and more integers).
If the number C is incompletely private, and the number D is the residue from dividing an integer A per integer B, then this fact we will briefly record as equality of the form A: B \u003d C (OST. D).
Note that when dividing an integer number A to an integer B, the residue may be zero. In this case, they say that A is divided into B without residue (or ncape). Thus, the division of integers without a residue is a special case of division of integers with the residue.
It is also worth saying that when dividing zero for some integer, we are always dealing with a division without a balance, since in this case the private will be zero (see section of the theory of zero division by an integer), and the residue will also be zero.
Determined with terminology and designations, we will now understand with the meaning of dividing integers with the remnant.
The division of a whole negative number A to a whole positive number B can also be given to the meaning. To do this, consider a whole negative number as debt. Imagine this situation. A debt that makes items must pay off the B person by making the same contribution. The absolute value of incomplete private C in this case will determine the amount of debt of each of these people, and the residue D will show how much items will remain after paying the debt. Let us give an example. Suppose 2 people should 7 apples. If we assume that each of them should be 4 apples, then after paying the debt, they will remain 1 apple. This situation corresponds to equality (-7): 2 \u003d -4 (OST. 1).
Division with the residue of an arbitrary integer A for a whole a negative number We will not give any point, but we will leave the right to exist.
Theorem on the divisibility of integers with the residue
When we talked about the division of natural numbers with the residue, they found out that divisible A, divider B, incomplete private C and the residue D are relating to the equality a \u003d b · c + d. For integers, A, B, C and D is characterized by the same connection. This link is approved by the following definition theorem with the residue.
Theorem.
Any integer A may be the only way through an integer and different from zero number B as a \u003d b · Q + R, where Q and R are some integers, and.
Evidence.
First, we prove the possibility of representation a \u003d b · Q + r.
If integers a and b such that A is divided into b aimed, then by definition there is such an integer q that a \u003d b · q. In this case, there is an equality a \u003d b · Q + R at r \u003d 0.
Now we assume that B is an integer positive number. Choose an integer q in such a way that the product B · Q does not exceed the number A, and the product B · (Q + 1) was already greater than a. That is, take q such that inequalities B · Q It remains to prove the possibility of representation a \u003d b · q + r for negative b. Since the module of the number B in this case is a positive number, then for the presentation, where Q 1 is some integer, and R is an integer satisfying conditions. Then, adopting q \u003d -q 1, we obtain the idea of \u200b\u200bthe visual representation A \u003d B · Q + R for the negative b. Go to the proof of uniqueness. Suppose that in addition to the representation A \u003d b · Q + R, Q and R - integers and, there is another representation A \u003d B · Q 1 + R 1, where Q 1 and R 1 are some integers, and Q 1 ≠ Q and. After subtracting from the left and right part of the first equality, respectively, the left and right part of the second equality, we obtain 0 \u003d b · (Q - Q 1) + R-R 1, which is equivalent to the equality R-R 1 \u003d b · (q 1 -q) . Then the equality of the species must be true From the conditions and it can be concluded that. As q and q 1 are integer and q ≠ q 1, then where we conclude that Equality A \u003d B · C + D allows you to find an unknown divide, if a divider B is known, incomplete private C and the residue d. Consider an example. Example. What is equally divisible if it is possible for an integer -21, an incomplete private 5 and a residue 12? Decision. We need to calculate Delimi A, when the divider B \u003d -21 is known, incomplete enough C \u003d 5 and the residue d \u003d 12. By contacting the equality A \u003d b · C + D, we obtain A \u003d (- 21) · 5 + 12. Observing, first, we first spend the multiplication of integers -21 and 5 according to the rule of multiplication of integers with different signs, after which we perform the addition of integers with different signs: (-21) · 5 + 12 \u003d -105 + 12 \u003d -93. Answer: −93
.
Relations between divisible, divisory, incomplete private and residue are also expressed by equalities of the form B \u003d (A - D): C, C \u003d (A - D): B and D \u003d A-B · C. These equalities allow calculating the divider, incomplete private and residue, respectively. We often have to find a residue from dividing an integer A to an integer B, when a divide, divider and incomplete private, using the formula d \u003d a-b · c. So that there are no questions in the future, we will analyze an example of calculating the residue. Example. Find the balance from dividing an integer -19 to an integer 3, if it is known that incomplete private equal to -7. Decision. To calculate the residue from division, we use the formula of the form d \u003d a - b · c. From the condition we have all the necessary data A \u003d -19, B \u003d 3, C \u003d -7. We get D \u003d A-B · C \u003d -19-3 · (-7) \u003d -19 - (- 21) \u003d - 19 + 21 \u003d 2 (difference -19 - (- 21) we calculated according to the rule of subtraction of a whole negative number ). Answer: As we have repeatedly noted, the whole positive numbers are natural numbers. Therefore, division with the residue of entire positive numbers is carried out in all rules of division with the residue of natural numbers. It is very important to be able to easily perform division with the residue of natural numbers, since it is the basis of dividing not only entire positive numbers, but also at the heart of all division rules with the residue of arbitrary integers. From our point of view, it is most convenient to perform division by a column, this method allows you to get and incomplete private (or just private) and the residue. Consider an example of division with the residue of entire positive numbers. Example. Perform a division with the residue of the number 14 671 by 54. Decision. Perform the division of these positive numbers by the Stage: Incomplete private turned out to be equal to 271, and the residue is 37. Answer: 14 671: 54 \u003d 271 (OST. 37). We formulate a rule that allows you to perform division with a whole positive number to a whole negative number. Incomplete private from division of an integer positive number A to a whole negative number B is a number opposite to incompletely private from division A to the module of the number B, and the residue from division A on B is equal to the balance of division by. This rule implies that incomplete private from dividing the integer positive number to a whole negative number is an integrity. We remake the announced rule in the division algorithm with the residue of a whole positive number of adequate: We give an example of using an algorithm for dividing a whole positive number to a whole negative. Example. Perform a division with the residue of a positive number 17 to a whole negative number -5. Decision. We use the division algorithm with the residue of a positive number to a whole negative. Sharing The number is the opposite of the number 3 is -3. Thus, the desired incomplete private from division 17 to -5 is -3, and the residue is 2. Answer: 17: (- 5) \u003d - 3 (OST. 2). Example. Divide 45 on -15. Decision. Delimo and divider modules are 45 and 15, respectively. The number 45 is divided into 15 without a residue, the private is equal to 3. Consequently, an integer positive number 45 is divided into a whole negative number -15 without a residue, the private at the same time is equal to the number opposite to 3, that is, -3. Indeed, according to the rule of division of integers with different signs we have. Answer: 45:(−15)=−3
.
We will give the wording the rules of division with the remnant of a whole negative number to a whole positive. In order to obtain an incomplete private C from dividing a whole negative number A to a whole positive number B, you need to take a number opposite to incompletely privately from the division of the modules of the initial numbers and deduct the unit from it, after which the residue D is calculated according to the formula d \u003d a-b · c. From this division rule with the residue, it follows that incomplete private from division of a whole negative for a whole positive number is a whole negative number. From the voiced rule implies the division algorithm with the balance of a whole negative number A to the whole positive B: We will analyze the solution of the example in which we use the recorded algorithm of division with the residue. Example. Find an incomplete private and residue from dividing a whole negative number -17 for a whole positive number 5. Decision. The dividera -17 module is 17, and the divider module 5 is 5. Sharing 17 to 5, we get incomplete private 3 and residue 2. The number opposite 3 is -3. We subtract from -3 unit: -3-1 \u003d -4. So, the desired incomplete private is -4. It remains to calculate the residue. In our example a \u003d -17, b \u003d 5, c \u003d -4, then d \u003d a-b · c \u003d -17-5 · (-4) \u003d -17 - (- 20) \u003d - 17 + 20 \u003d 3 . Thus, the incomplete private from division of a whole negative number -17 to an integer positive number 5 is -4, and the residue is 3. Answer: (-17): 5 \u003d -4 (OST. 3). Example. Divide the whole negative number -1 404 by a positive number 26. Decision. The dividend module is 1 404, the divider module is 26. We split 1 404 on the 26th Stage: Since the divide module was divided into a divider module without a residue, the initial integers are divided without a residue, and the desired private is equal to the number opposite to 54, that is, -54. Answer: (−1 404):26=−54
.
We formulate a division rule with the residue of whole negative numbers. In order to obtain an incomplete private C from dividing a whole negative number A to a whole negative number B, it is necessary to calculate the incomplete private on the division of the initial numbers modules and add a unit to it, after that the residue D calculates according to the formula d \u003d a-b · c. This rule implies that incomplete private from division of entire negative numbers is a whole positive number. We rewrite the voiced rule in the form of an algorithm for dividing whole negative numbers: Consider the use of the algorithm for dividing whole negative numbers when solving an example. Example. Find an incomplete private and residue from dividing a whole negative number -17 to a whole negative number -5. Decision. We use the corresponding algorithm of division with the residue. The dividend module is 17, the divider module is 5. Division 17 on 5 gives incomplete private 3 and residue 2. By incomplete private 3 add unit: 3 + 1 \u003d 4. Consequently, the desired incomplete private from division -17 to -5 is 4. It remains to calculate the residue. In this example, a \u003d -17, b \u003d -5, c \u003d 4, then d \u003d a-b · c \u003d -17 - (- 5) · 4 \u003d -17 - (- 20) \u003d - 17 + 20 \u003d 3 . So, the incomplete private from dividing a whole negative number -17 to a whole negative number -5 is 4, and the residue is 3. Answer: (-17): (- 5) \u003d 4 (OST. 3). After the determination of integers with the residue is made, it is useful to check the result obtained. Check is carried out in two stages. At the first stage is checked whether the residue D is a non-negative number, and the condition is checked. If all the conditions of the first stage of the check are made, then you can begin to the second stage of checking, otherwise it can be argued that an error was made when dividing with the residue. At the second stage, the validity of the equality A \u003d B · C + D is checked. If this equality is valid, the division with the residue was carried out correctly, otherwise an error was made somewhere. Consider solutions of examples in which the result of dividing the integers with the residue is performed. Example. When dividing the number -521 on -12, incomplete private 44 and residue 7 were obtained, follow the result. Decision. -2 for b \u003d -3, c \u003d 7, d \u003d 1. Have b · C + d \u003d -3 · 7 + 1 \u003d -21 + 1 \u003d -20. Thus, the equality A \u003d B · C + D is incorrect (in our example a \u003d -19). Consequently, the division with the residue was incorrect. The article dissellites the concept of dividing integers with the residue. We prove the theorem on the divisibility of integers with the residue and view the relationship between divisions and divisors, incomplete private and residues. Consider the rules when the whole numbers are divided with the remnants, examined in detail on the examples. At the end of the decision will perform a check. The division of integers with the residue is considered as a generalized division with the residue of natural numbers. This is done because natural numbers are an integral part of the whole. The division with the residue of an arbitrary number suggests that an integer A is divided by the number B, different from zero. If b \u003d 0, then do not produce a division with the residue. As well as the division of natural numbers with the residue, the division of integers A and B is made, with B different from zero, on C and d. In this case, A and B are called divisible and divider, and D is the balance residue, C is an integer or incomplete private. If we assume that the residue is a non-negative number, then its value is not larger than the number b. We write in this way: 0 ≤ d ≤ b. This chain of inequalities is used when comparing 3 and more than the number of numbers. If C is an incomplete private, then D is the residue from dividing an integer A per B, briefly can be fixed: A: B \u003d C (OST. D). The residue during the division of the numbers A on b is possible zero, then they say that A is divided on b a focus, that is, without a residue. Division without a residue is considered a special case of division. If we divide zero for some number, we obtain as a result of zero. The balance residue will also be zero. This can be traced from the theory of dividing zero by an integer. Now consider the meaning of dividing integers with the residue. It is known that the whole positive numbers are natural, then when dividing with the residue, it will be the same sense, as in the division of natural numbers with the residue. When dividing a whole negative number A, a whole positive b there is a meaning. Consider on the example. Representing the situation when we have a debt of objects in the amount of A, which you need to pay off B. To do this, you need to make the same contribution to everyone. To determine the amount of debt for everyone, it is necessary to pay attention to the size of the private with. The residue D says that a number of items are known after a disclaimer with debts. Consider on the example with apples. If 2 people should 7 apples. In case it is considered that everyone must return to 4 apples, after a complete calculation, they will remain 1 apple. We write in the form of equality it: (- 7): 2 \u003d - 4 (o with t. 1). The division of any number and does not make sense, but perhaps as an option. We revealed that a - this is divisible, then B is a divider, with - incomplete private, and D is the residue. They are connected with each other. This connection will show with the help of equality a \u003d b · c + d. The relationship between them is characterized by the theorest division with the residue. Theorem Any integer can be represented only through an integer and different from zero number B in this way: a \u003d b · Q + R, where Q and R are some integers. Here we have 0 ≤ r ≤ b. We prove the possibility of existence A \u003d B · Q + R. Evidence If there are two numbers a and b, and A is divided into B without residue, then it follows from the definition that there is a number Q, which will be true equality A \u003d B · Q. Then the equality can be considered true: a \u003d b · Q + R with R \u003d 0. Then it is necessary to take Q such that this inequality b · q< a < b · (q + 1) было верным. Необходимо вычесть b · q из всех частей выражения. Тогда придем к неравенству такого вида: 0 < a − b · q < b . We have that the value of the expression A - b · Q is greater than zero and no more value of the number B, it follows that R \u003d a - b · Q. We obtain that the number A can be represented as a \u003d b · Q + r. Now it is necessary to consider the possibility of representation a \u003d b · Q + R for negative values \u200b\u200bb. The module of the number is obtained positive, then we obtain a \u003d b · q 1 + r, where the value Q 1 is some integer, R is an integer that suits the condition 0 ≤ R< b . Принимаем q = − q 1 , получим, что a = b · q + r для отрицательных b . Proof of uniqueness Suppose that A \u003d b · Q + R, Q and R are integers with a faithful condition 0 ≤ R< b , имеется еще одна форма записи в виде a = b · q 1 + r 1 , где Q 1. and R 1 are some numbers where q 1 ≠ q 0 ≤ R 1< b . When inequality is subtracted from the left and right parts, then we obtain 0 \u003d b · (Q - Q 1) + R 1, which is equivalent to R - R 1 \u003d b · Q 1 - Q. Since the module is used, we obtain the equality R - R 1 \u003d b · Q 1 - Q. The specified condition suggests that 0 ≤ R< b и 0 ≤ r 1 < b запишется в виде r - r 1 < b . Имеем, что Q.and Q 1.- whole, and Q ≠ Q 1, then Q 1 - Q ≥ 1. From here we have that b · q 1 - q ≥ b. The obtained inequalities R - R 1< b и b · q 1 - q ≥ b указывают на то, что такое равенство в виде r - r 1 = b · q 1 - q невозможно в данном случае. It follows that a different number A is presented can not be presented, except as such a record A \u003d B · Q + R. With the help of the equality A \u003d b · C + D, an unknown division can be found when a divider B is known with an incomplete private C and the residue d. Example 1. Determine dividimi if the division is obtained - 21, incomplete private 5 and residue 12. Decision
It is necessary to calculate Delimi A with the known divider B \u003d - 21, incomplete private C \u003d 5 and the residue d \u003d 12. It is necessary to refer to the equality A \u003d b · C + D, we obtain a \u003d (- 21) · 5 + 12. Under the procedure for performing actions, multiply - 21 to 5, after that we obtain (- 21) · 5 + 12 \u003d - 105 + 12 \u003d - 93. Answer: - 93 . The relationship between the divider and incomplete private and the residue can be expressed using equations: B \u003d (a - d): C, C \u003d (A - D): B and d \u003d a - b · c. With their help, we can calculate the divider, incomplete private and residue. This reduces to the constant finding of the residue from dividing the whole integers A on B with a known divisible, divider and incomplete private. The formula d \u003d a - b · c is applied. Consider the decision in detail. Example 2. Find the residue from the division of an integer - 19 by a whole 3 with a known incomplete private equal to 7. Decision
To calculate the residue from the division, we apply the formula of the form d \u003d a - b · c. By condition, all data A \u003d - 19, B \u003d 3, C \u003d - 7 are available. From here we obtain D \u003d A - B · C \u003d - 19 - 3 · (- 7) \u003d - 19 - (- 21) \u003d - 19 + 21 \u003d 2 (the difference is 19 - (- 21). This example is calculated according to the deduction rule. A whole negative number. Answer: 2 . All integer positive numbers are natural. It follows that division is performed on all division rules with the residue of natural numbers. The rate of execution of division with the residue of natural numbers is important, since it is founded not only the division of positive, but also the rules for dividing whole arbitrary. The most convenient method of division is a column, as it is easier and faster to get incomplete or just a private with the residue. Consider the decision in more detail. Example 3. Decision 14671 to 54. Decision
This division must be performed by a column: That is, incomplete private is obtained equal to 271, and the residue is 37. Answer: 14 671: 54 \u003d 271. (OST. 37) To divide with the residue of a positive number for a whole negative, it is necessary to formulate a rule. Definition 1. Incomplete private from division of a whole positive A to a whole negative B receive a number that is opposite to incompletely private from dividing the numbers A per b. Then the residue is equal to the residue when dividing A on b. From here we have that incompletely private from division of a whole one-time number for a whole negative number is considered to be an integer non-mental number. We obtain the algorithm: Consider on the example of the algorithm for dividing a whole positive number to a whole negative. Example 4. Perform division with the residue 17 to 5. Decision
Apply a division algorithm with a whole positive number of a whole negative. It is necessary to divide the 17 to - 5 module. From here we get that incomplete private is 3, and the residue is 2. We obtain that the desired number from division 17 to - 5 \u003d - 3 with the residue is equal to 2. Answer: 17: (- 5) \u003d - 3 (OST. 2). Example 5. It is necessary to divide 45 to 15. Decision
It is necessary to divide the numbers by the module. The number 45 is divided by 15, we will get a private 3 without a residue. So, the number 45 is divided into 15 without a residue. In response, we get - 3, since the division was performed in the module. 45: (- 15) = 45: - 15 = - 45: 15 = - 3 Answer: 45: (− 15) = − 3 . The wording of division rules with the residue is as follows. Definition 2. In order to obtain an incomplete private C when dividing a whole negative A per positive b, you need to apply the opposite to this number and subtract from it 1, then the residue D will be calculated by the formula: d \u003d a - b · c. Based on the rule, it can be concluded that when dividing, we obtain a non-negative number. For the accuracy of the solution, the algorithm of division A on B is used with the residue: Consider on the example of a solution where this algorithm applies. Example 6. Find an incomplete private and balance from division - 17 to 5. Decision
We divide the specified numbers in the module. We obtain that in the division of the private equal to 3, and the remainder 2. Since they got 3, opposite - 3. It is necessary to take away 1. − 3 − 1 = − 4 . The desired value is 100th equal to 4. To calculate the residue, it is necessary a \u003d - 17, b \u003d 5, c \u003d - 4, then d \u003d a - b · c \u003d - 17 - 5 · (- 4) \u003d - 17 - (- 20) \u003d - 17 + 20 \u003d 3. So, the incomplete private from the division is the number - 4 with the residue equal to 3. Answer: (- 17): 5 \u003d - 4 (OST. 3). Example 7. Split a whole negative number - 1404 per positive 26. Decision
It is necessary to divide the column and on Mudlyuly. We got the division of the modules of numbers without a residue. This means that the division is performed without a residue, but the artistic private \u003d - 54. Answer: (− 1 404) : 26 = − 54 . It is necessary to formulate a division rule with the residue of entire negative numbers. Definition 3. To obtain an incomplete private C from dividing a whole negative number A to a whole negative b, it is necessary to calculate the module in the module, after which add 1, then we can make calculations according to the formula d \u003d a - b · c. From here it follows that the incomplete private from division of entire negative numbers will be the number is positive. We formulate this rule as an algorithm: This algorithm will look at the example. Example 8. Find an incomplete private and residue during division - 17 to 5. Decision
For the correctness of the decision, we apply an algorithm for dividing with the residue. To begin withdrawn the number in the module. From here we get that incomplete private \u003d 3, and the residue is 2. According to the rule, it is necessary to add incomplete private and 1. We obtain that 3 + 1 \u003d 4. From here we get that the incomplete private from the division of the given numbers is 4. To calculate the residue, we apply the formula. By condition, we have that a \u003d - 17, b \u003d - 5, c \u003d 4, then, using the formula, we obtain D \u003d A - B · C \u003d - 17 - (- 5) · 4 \u003d - 17 - (- 20) \u003d - 17 + 20 \u003d 3. The desired answer, that is, the residue is 3, and the incomplete private is 4. Answer: (- 17): (- 5) \u003d 4 (OST. 3). After making the division of numbers with the residue, you must check. This check implies 2 stages. Initially, there is a check of the residue D to non-negativity, the performance of the condition 0 ≤ D< b . При их выполнении разрешено выполнять 2 этап. Если 1 этап не выполнился, значит вычисления произведены с ошибками. Второй этап состоит из того, что равенство a = b · c + d должно быть верным. Иначе в вычисления имеется ошибка. Consider at the examples. Example 9. The division was produced - 521 on - 12. Private equal to 44, residue 7. Perform check. Decision
Since the residue is a positive number, then its value is less than the divisor module. The divider is equal to 12, it means that its module is 12. You can go to the next check item. By condition, we have that a \u003d - 521, b \u003d - 12, c \u003d 44, d \u003d 7. From here, we calculate B · C + D, where b · C + d \u003d - 12 · 44 + 7 \u003d - 528 + 7 \u003d - 521. It follows that the equality is correct. Check is passed. Example 10. Check the division (- 17): 5 \u003d - 3 (OST. - 2). Is equality true? Decision
The meaning of the first stage is that it is necessary to check the division of integers with the residue. It can be seen that the action is made incorrectly, since the residue is equal to 2. The residue is not a negative number. We have that the second condition is made, but not sufficient for this case. Answer: not. Example 11. The number - 19 was divided into 3. Incomplete private equal to 7, and residue 1. Check if this calculation is true. Decision
Dan a residue equal to 1. He is positive. By magnitude less than the divider module, it means that the first stage is performed. Let us turn to the second stage. Calculate the value of the expression B · C + D. By condition, we have that b \u003d - 3, c \u003d 7, d \u003d 1, it means that substituting the numeric values, we obtain b · C + d \u003d - 3 · 7 + 1 \u003d - 21 + 1 \u003d - 20. It follows that a \u003d b · C + D equality is not performed, since the condition is given a \u003d - 19. Hence the conclusion that the division is made with an error. Answer: not. If you notice a mistake in the text, please select it and press Ctrl + Enter Consider a simple example: Sometimes the natural number is completely able to divide the focus. For example, consider the task: Decision: We know that 16 is not to share. The nearmost number that is divided by 5 is 15 and 1 in the remainder. Number 15 We can paint as 5⋅3. As a result (16 - Delimi, 5 - divider, 3 - incomplete private, 1 - residue). Received formula division with the residuewhich can be done solution check. a.=
b.⋅
c.+
d. Answer: Every child will take 3 toys and one toy will remain. The residue should always be less than the divider. If when dividing the residue is zero, it means that divisible sharing ncape Or without a balance on the divider. If when dividing the residue is more divisor, it means that the found number is not the biggest. There is a larger number that divide and the residue will be less than a divider. Questions on the topic "Decision with the residue":
The residue can be equal to the divider? How to find divisible on incomplete private, divider and residue? Example number 1:
Decision: 258 - Delimi, b) We divide the column: 1873 - Delimi, Substitute in the formula and check whether we decided to solve the example: Example number 2:
Answer: Example number 3:
Answer: Example number 4:
Decision: b) resolved with the help of formula: A task:
Wire 4m. It is necessary to cut into pieces of 13cm. How many such pieces will it work? Decision: Answer: 30 pieces turn out and 10 cm. Wire will remain.
, and by virtue of the properties of the module of the number - and equality 
.
. From the obtained inequalities and It follows that equality of the form It is impossible at our assumption. Therefore, there is no other representation of the number A, except a \u003d b · Q + r.Links between divisible, divider, incomplete private and residue
Division with the residue of entire positive numbers, examples
The division rule with the residue of a positive number of adequate, examples
Division with a whole negative number of integer positive, examples
The division rule with the residue of whole negative numbers, examples
Check the result of dividing integers with the residue
General view of division of integers with residues
Theorem on the divisibility of integers with the residue
Communication between divisible, divider, incomplete private and residue
The division rule with the residue of a positive number of adequate, examples
The division rule with the residue of whole negative numbers, examples
Check the result of dividing integers with the residue
15:5=3
In this example, the natural number of 15 we divided ncape3, without a balance.
16 toys lay in the closet. The group had five children. Each child took the same number of toys. How many toys have every child?
We divide the number 16 on 5 column we get:
a. - Delimi,
b. - divider,
c. - incomplete private,
d. - Balance.Remainder of the division
The remainder may be more divider?
Answer: No.
Answer: No.
Answer: The values \u200b\u200bof the incomplete private, divider and the residue are substituted into the formula and find divisible. Formula:
a \u003d b⋅c + d
Perform a division with the residue and check: a) 258: 7 b) 1873: 8
a) We divide the column: 
7 - divider,
36 - incomplete private,
6 - residue. Residue less divider 6<7.
7⋅36+6=252+6=258
8 - divider,
234 - incomplete private,
1 - residue. The residue is less than divider 1<8.
8⋅234+1=1872+1=1873
What remnants are obtained when dividing natural numbers: a) 3 b) 8?
a) The residue is less than the divider, therefore, less 3. In our case, the residue can be equal to 0, 1 or 2.
b) The residue is less than the divider, therefore, less than 8. In our case, the residue can be equal to 0, 1, 2, 3, 4, 5, 6 or 7.
What the greatest residue may turn out when dividing natural numbers: a) 9 b) 15?
a) the residue is less than the divider, therefore, less than 9. But we need to specify the greatest balance. That is the nearest number to the divider. This is the number 8.
b) the residue is less than the divider, therefore, less than 15. But we need to specify the greatest balance. That is the nearest number to the divider. This is the number 14.
Find divisible: a) A: 6 \u003d 3 (OST 4) b) C: 24 \u003d 4 (East.11)
a) soluing with the help of formula:
a \u003d b⋅c + d
(A - Delimi, B - divider, C - incomplete private, D - residue.)
A: 6 \u003d 3 (OST.4)
(A - Delimi, 6 - divider, 3 - incomplete private, 4 - residue.) Substitute the numbers in the formula:
a \u003d 6⋅3 + 4 \u003d 22
Answer: A \u003d 22
a \u003d b⋅c + d
(A - Delimi, B - divider, C - incomplete private, D - residue.)
C: 24 \u003d 4 (East.11)
(C - Delimi, 24 - divider, 4 - incomplete private, 11 - residue.) Substitute the numbers in the formula:
C \u003d 24⋅4 + 11 \u003d 107
Answer: C \u003d 107
First you need to translate meters to centimeters.
4m. \u003d 400cm.
You can share a column or in the mind we will get:
400: 13 \u003d 30 (OST.10)
Check:
13⋅30+10=390+10=400
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