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Determine whether each number is a multiple of 5 .

  1. 675
  2. 1,578

  1. yes
  2. no

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Determine whether each number is a multiple of 5 .

  1. 421
  2. 2,690

  1. no
  2. yes

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[link] highlights the multiples of 10 between 1 and 50 . All multiples of 10 all end with a zero.

The image shows a chart with five rows and ten columns. The first row lists the numbers from 1 to 10. The second row lists the numbers from 11 to 20. The third row lists the numbers from 21 to 30. The fourth row lists the numbers from 31 and 40. The fifth row lists the numbers from 41 to 50. All factors of 10 are highlighted in blue.
Multiples of 10 between 1 and 50

Determine whether each of the following is a multiple of 10 :

  1. 425
  2. 350

Solution

Is 425 a multiple of 10?
Is the last digit zero? No.
425 is not a multiple of 10.
Is 350 a multiple of 10?
Is the last digit zero? Yes.
350 is a multiple of 10.
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Determine whether each number is a multiple of 10 :

  1. 179
  2. 3,540

  1. no
  2. yes

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Determine whether each number is a multiple of 10 :

  1. 110
  2. 7,595

  1. yes
  2. no

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[link] highlights multiples of 3 . The pattern for multiples of 3 is not as obvious as the patterns for multiples of 2 , 5 , and 10 .

The image shows a chart with five rows and ten columns. The first row lists the numbers from 1 to 10. The second row lists the numbers from 11 to 20. The third row lists the numbers from 21 to 30. The fourth row lists the numbers from 31 and 40. The fifth row lists the numbers from 41 to 50. All factors of 3 are highlighted in blue.
Multiples of 3 between 1 and 50

Unlike the other patterns we’ve examined so far, this pattern does not involve the last digit. The pattern for multiples of 3 is based on the sum of the digits. If the sum of the digits of a number is a multiple of 3 , then the number itself is a multiple of 3 . See [link] .

Multiple of 3 3 6 9 12 15 18 21 24
Sum of digits 3 6 9 1 + 2 3 1 + 5 6 1 + 8 9 2 + 1 3 2 + 4 6

Consider the number 42 . The digits are 4 and 2 , and their sum is 4 + 2 = 6 . Since 6 is a multiple of 3 , we know that 42 is also a multiple of 3 .

Determine whether each of the given numbers is a multiple of 3 :

  1. 645
  2. 10,519

Solution

Is 645 a multiple of 3 ?

Find the sum of the digits. 6 + 4 + 5 = 15
Is 15 a multiple of 3? Yes.
If we're not sure, we could add its digits to find out. We can check it by dividing 645 by 3. 645 ÷ 3
The quotient is 215. 3 215 = 645

Is 10,519 a multiple of 3 ?

Find the sum of the digits. 1 + 0 + 5 + 1 + 9 = 16
Is 16 a multiple of 3? No.
So 10,519 is not a multiple of 3 either.. 645 ÷ 3
We can check this by dividing by 10,519 by 3. 3,506 R 1 3 10,519

When we divide 10,519 by 3 , we do not get a counting number, so 10,519 is not the product of a counting number and 3 . It is not a multiple of 3 .

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Determine whether each number is a multiple of 3 :

  1. 954
  2. 3,742

  1. yes
  2. no

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Determine whether each number is a multiple of 3 :

  1. 643
  2. 8,379

  1. no
  2. yes

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Look back at the charts where you highlighted the multiples of 2 , of 5 , and of 10 . Notice that the multiples of 10 are the numbers that are multiples of both 2 and 5 . That is because 10 = 2 5 . Likewise, since 6 = 2 3 , the multiples of 6 are the numbers that are multiples of both 2 and 3 .

Use common divisibility tests

Another way to say that 375 is a multiple of 5 is to say that 375 is divisible by 5 . In fact, 375 ÷ 5 is 75 , so 375 is 5 75 . Notice in [link] that 10,519 is not a multiple 3 . When we divided 10,519 by 3 we did not get a counting number, so 10,519 is not divisible by 3 .

Divisibility

If a number m is a multiple of n , then we say that m is divisible by n .

Since multiplication and division are inverse operations, the patterns of multiples that we found can be used as divisibility tests. [link] summarizes divisibility tests for some of the counting numbers between one and ten.

Divisibility Tests
A number is divisible by
2 if the last digit is 0 , 2 , 4 , 6 , or 8
3 if the sum of the digits is divisible by 3
5 if the last digit is 5 or 0
6 if divisible by both 2 and 3
10 if the last digit is 0

Determine whether 1,290 is divisible by 2 , 3 , 5 , and 10 .

Solution

[link] applies the divisibility tests to 1,290 . In the far right column, we check the results of the divisibility tests by seeing if the quotient is a whole number.

Divisible by…? Test Divisible? Check
2 Is last digit 0 , 2 , 4 , 6 , or 8 ? Yes. yes 1290 ÷ 2 = 645
3 Is sum of digits divisible by 3 ?
1 + 2 + 9 + 0 = 12 Yes.
yes 1290 ÷ 3 = 430
5 Is last digit is 5 or 0 ? Yes. yes 1290 ÷ 5 = 258
10 Is last digit 0 ? Yes. yes 1290 ÷ 10 = 129

Thus, 1,290 is divisible by 2 , 3 , 5 , and 10 .

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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