Topic 8 of 12

Factors and multiples

Step 8 · Ratios and negative numbers topics

Factors divide a number exactly; multiples are what you get by counting up in steps of it. Learn the greatest common factor, the least common multiple, and how the distributive property uses them.

Factors

A factor of a number divides it exactly, with nothing left over. Since $4 \times 9 = 36$ , both 4 and 9 are factors of 36. To find every factor, look for pairs that multiply to the number:

36 = 1 \times 36 = 2 \times 18 = 3 \times 12 = 4 \times 9 = 6 \times 6

So the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. Every number has 1 and itself as factors.

Multiples

A multiple of a number is what you get when you multiply it by 1, 2, 3 and so on. The multiples of 6 are 6, 12, 18, 24, 30, and the list never ends. Factors are the numbers that go into a number; multiples are the numbers it goes into.

Prime and composite numbers

A prime number has exactly two factors: 1 and itself. 2, 3, 5, 7, 11 and 13 are prime. A number with more than two factors is composite, like 12. The number 1 has only one factor, so it is neither prime nor composite.

Every composite number can be written as a product of primes in just one way (apart from the order). Splitting 60 up step by step gives its prime factorization:

60 = 6 \times 10 = (2 \times 3) \times (2 \times 5) = 2 \times 2 \times 3 \times 5

Greatest common factor

A common factor of two numbers divides both of them. The greatest common factor (GCF) is the largest one.

Example: The GCF of 36 and 48

Find the greatest common factor of 36 and 48.

List the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
List the factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
The numbers in both lists are 1, 2, 3, 4, 6 and 12. The largest is 12.

AnswerThe GCF is 12.

Example: Equal bags

A teacher has 24 pencils and 36 erasers. She wants to make identical bags, using everything, with as many bags as possible. How many bags can she make, and what is in each?

The number of bags must divide both 24 and 36, so it is a common factor.
The most bags means the greatest common factor. The GCF of 24 and 36 is 12.
Each bag gets $24 \div 12 = 2$ pencils and $36 \div 12 = 3$ erasers.

Answer12 bags, each with 2 pencils and 3 erasers.

Least common multiple

A common multiple of two numbers is a multiple of both. The least common multiple (LCM) is the smallest one. For 4 and 6, the multiples of 4 are 4, 8, 12, 16 and the multiples of 6 are 6, 12, 18. The first number in both lists is 12.

Example: Two buses

One bus leaves the station every 6 minutes and another every 8 minutes. They leave together at 8:00. When do they next leave together?

Both times are multiples of 6 and of 8, so we want a common multiple, and the next time means the least one.
Multiples of 6: 6, 12, 18, 24, 30. Multiples of 8: 8, 16, 24, 32.
The first match is 24.

Answer24 minutes later, at 8:24.

Using prime factors

For big numbers, prime factorizations are quicker than lists. Write $36 = 2^2 \times 3^2$ and $48 = 2^4 \times 3$ . The GCF takes the smaller power of each prime and the LCM takes the larger:

\text{GCF} = 2^2 \times 3 = 12 \qquad \text{LCM} = 2^4 \times 3^2 = 144

The distributive property

The distributive property says $a(b + c) = ab + ac$ . Read backwards, it lets you pull a common factor out of a sum. Use the GCF to do it fully:

36 + 48 = 12 \times 3 + 12 \times 4 = 12 \times (3 + 4) = 12 \times 7 = 84

Try it yourself

Work each one on paper first, then open the answer to check.

List all the factors of 24.
Show answer
1, 2, 3, 4, 6, 8, 12 and 24.
Find the GCF of 18 and 30.
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6. The factors of 18 are 1, 2, 3, 6, 9, 18 and of 30 are 1, 2, 3, 5, 6, 10, 15, 30, so the greatest one in both is 6.
Find the LCM of 9 and 12.
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36. The multiples of 12 are 12, 24, 36, and 36 is the first that 9 divides into ( $9 \times 4 = 36$ ).
Write 84 as a product of primes.
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$2 \times 2 \times 3 \times 7$ , because $84 = 4 \times 21 = 2 \times 2 \times 3 \times 7$ .
Use the distributive property to write $28 + 42$ as a GCF times a sum, and find the total.
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$14 \times (2 + 3) = 14 \times 5 = 70$ . The GCF of 28 and 42 is 14.
Two lights blink every 10 seconds and every 15 seconds. They blink together now. When will they next blink together?
Show answer
In 30 seconds, because the LCM of 10 and 15 is 30.