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 into 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 into 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 into 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 any of the natural numbers can be selected its individual sign of divisibility or the "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.
Consider a simple example:
15:5=3
In this example natural number 15 We are divided ncape3, without a balance.
Sometimes the natural number is completely able to divide the focus. For example, consider the task:
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?
Decision:
We divide the number 16 on 5 column we get:
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.
a. - Delimi,
b. - divider,
c. - incomplete private,
d. - Balance.
Answer: Every child will take 3 toys and one toy will remain.
Remainder of the division
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 remainder may be more divider?
Answer: No.
The residue can be equal to the divider?
Answer: No.
How to find divisible on incomplete private, divider and residue?
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
Example number 1:
Perform a division with the residue and check: a) 258: 7 b) 1873: 8
Decision:
a) We divide the column:
258 - Delimi,
7 - divider,
36 - incomplete private,
6 - residue. Residue less divider 6<7.
7⋅36+6=252+6=258
b) We divide the column:
1873 - Delimi,
8 - divider,
234 - incomplete private,
1 - residue. The residue is less than divider 1<8.
Substitute in the formula and check whether we decided to solve the example:
8⋅234+1=1872+1=1873
Example number 2:
What remnants are obtained when dividing natural numbers: a) 3 b) 8?
Answer:
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.
Example number 3:
What the greatest residue may turn out when dividing natural numbers: a) 9 b) 15?
Answer:
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.
Example number 4:
Find divisible: a) A: 6 \u003d 3 (OST 4) b) C: 24 \u003d 4 (East.11)
Decision:
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
b) resolved with the help of formula:
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
A task:
Wire 4m. It is necessary to cut into pieces of 13cm. How many such pieces will it work?
Decision:
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
Answer: 30 pieces turn out and 10 cm. Wire will remain.
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 an integral part of the 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).
A division with the residue of an arbitrary integer A for a whole 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.