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Determine whether the given number is divisible by 2 , 3 , 5 , and 10 .

6240

Divisible by 2, 3, 5, and 10

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Determine whether the given number is divisible by 2 , 3 , 5 , and 10 .

7248

Divisible by 2 and 3, not 5 or 10.

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Determine whether 5,625 is divisible by 2 , 3 , 5 , and 10 .

Solution

[link] applies the divisibility tests to 5,625 and tests the results by finding the quotients.

Divisible by…? Test Divisible? Check
2 Is last digit 0 , 2 , 4 , 6 , or 8 ? No. no 5625 ÷ 2 = 2812.5
3 Is sum of digits divisible by 3 ?
5 + 6 + 2 + 5 = 18 Yes.
yes 5625 ÷ 5 = 1875
5 Is last digit is 5 or 0 ? Yes. yes 5625 ÷ 5 = 1125
10 Is last digit 0 ? No. no 5625 ÷ 10 = 562.5

Thus, 5,625 is divisible by 3 and 5 , but not 2 , or 10 .

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Determine whether the given number is divisible by 2 , 3 , 5 , and 10 .

4962

Divisible by 2, 3, and 6.

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Determine whether the given number is divisible by 2 , 3 , 5 , and 10 .

3765

Divisible by 3 and 5.

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Find all the factors of a number

There are often several ways to talk about the same idea. So far, we’ve seen that if m is a multiple of n , we can say that m is divisible by n . We know that 72 is the product of 8 and 9 , so we can say 72 is a multiple of 8 and 72 is a multiple of 9 . We can also say 72 is divisible by 8 and by 9 . Another way to talk about this is to say that 8 and 9 are factors of 72 . When we write 72 = 8 9 we can say that we have factored 72 .

The image shows the equation 8 times 9 equals 72. The 8 and 9 are labeled as factors and the 72 is labeled product.

Factors

If a b = m , then a and b are factors of m , and m is the product of a and b .

In algebra, it can be useful to determine all of the factors of a number. This is called factoring a number, and it can help us solve many kinds of problems.

Doing the Manipulative Mathematics activity “Model Multiplication and Factoring” will help you develop a better understanding of multiplication and factoring.

For example, suppose a choreographer is planning a dance for a ballet recital. There are 24 dancers, and for a certain scene, the choreographer wants to arrange the dancers in groups of equal sizes on stage.

In how many ways can the dancers be put into groups of equal size? Answering this question is the same as identifying the factors of 24 . [link] summarizes the different ways that the choreographer can arrange the dancers.

Number of Groups Dancers per Group Total Dancers
1 24 1 24 = 24
2 12 2 12 = 24
3 8 3 8 = 24
4 6 4 6 = 24
6 4 6 4 = 24
8 3 8 3 = 24
12 2 12 2 = 24
24 1 24 1 = 24

What patterns do you see in [link] ? Did you notice that the number of groups times the number of dancers per group is always 24 ? This makes sense, since there are always 24 dancers.

You may notice another pattern if you look carefully at the first two columns. These two columns contain the exact same set of numbers—but in reverse order. They are mirrors of one another, and in fact, both columns list all of the factors of 24 , which are:

1 , 2 , 3 , 4 , 6 , 8 , 12 , 24

We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with 1 . If the quotient    is also a counting number, then the divisor and the quotient are factors of the number. We can stop when the quotient becomes smaller than the divisor.

Find all the factors of a counting number.

  1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
    • If the quotient is a counting number, the divisor and quotient are a pair of factors.
    • If the quotient is not a counting number, the divisor is not a factor.
  2. List all the factor pairs.
  3. Write all the factors in order from smallest to largest.
Practice Key Terms 4

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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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