第1章基礎

The numbers 2, 4, 6, 8, 10, and 12 are called multiples of 2. A multiple of 2 can be written as the product of 2 and a counting number.

Similarly, a multiple of 3 would be the product of a counting number and 3.

We could find the multiples of any number by continuing this process.

Table 1.1 shows the multiples of 2 through 9 for the first 12 counting numbers.

Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 15 by 3, we get a counting number. In fact, $15÷3$ is 5, so 15 is $5·3.$

Look at the multiples of 5 in Table 1.1. They all end in 5 or 0. Numbers with last digit of 5 or 0 are divisible by 5. Looking for other patterns in Table 1.1 that shows multiples of the numbers 2 through 9, we can discover the following divisibility tests:

In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.

Since $8·9=72,$ we say that 8 and 9 are factors of 72. When we write $72=8·9,$ we say we have factored 72.

Other ways to factor 72 are $1·72,2·36,3·24,4·18,\text{and}6·12.$ Seventy-two has many factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, and 72.

Some numbers, like 72, have many factors. Other numbers have only two factors.

The counting numbers from 2 to 19 are listed in Figure 1.5, with their factors. Make sure to agree with the “prime” or “composite” label for each!

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.

A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful later in this course.

To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime!

If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators. Two methods are used most often to find the least common multiple and we will look at both of them.

The first method is the Listing Multiples Method. To find the least common multiple of 12 and 18, we list the first few multiples of 12 and 18:

Notice that some numbers appear in both lists. They are the common multiples of 12 and 18.

We see that the first few common multiples of 12 and 18 are 36, 72, and 108. Since 36 is the smallest of the common multiples, we call it the least common multiple. We often use the abbreviation LCM.

The procedure box lists the steps to take to find the LCM using the prime factors method we used above for 12 and 18.

Our second method to find the least common multiple of two numbers is to use The Prime Factors Method. Let’s find the LCM of 12 and 18 again, this time using their prime factors.

Notice that the prime factors of 12 $\left(2·2·3\right)$ and the prime factors of 18 $\left(2·3·3\right)$ are included in the LCM $(2·2·3·3).$ So 36 is the least common multiple of 12 and 18.

By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.