第2章代数の言葉

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 $5\cdot 75.$ Notice in Example 2.43 that $\mathrm{10,519}$ is not a multiple $3.$ When we divided $\mathrm{10,519}$ by $3$ we did not get a counting number, so $\mathrm{10,519}$ is not divisible by $3.$

Since multiplication and division are inverse operations, the patterns of multiples that we found can be used as divisibility tests. Table 2.11 summarizes divisibility tests for some of the counting numbers between one and ten.

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 $72=8\cdot 9$ we can say that we have factored $72.$

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.

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 $24.$ Table 2.14 summarizes the different ways that the choreographer can arrange the dancers.

What patterns do you see in Table 2.14? 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:

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.