Factors and Multiples

The factors of a number are integers that can be divided evenly (without a remainder) into the number.

For example, the number 20 is divisible by 5. This means that 5 is a factor of 20 because 5 can go into 20 four times without going over or under the number 20. Likewise, 4 is a factor of 20 because it can go into 20 evenly five times. However, the number 20 is not evenly divisible by 6, as 6 can only go into the number 20 three times before going over! In other words, the integers that are multiplied together to form a product are the factors of that product.

Common Factors

Common factors are factors that two or more numbers have in common, such as 4, which is a common factor of 16 and 20.

The greatest common factor of two or more numbers is the largest common factor shared between numbers, such as how 7 is the greatest common factor of 14 and 21.

Prime Factors

Prime factors are the factors of a number that are prime numbers. 1 is NOT a prime factor because it is not a prime number.

Multiples

The multiples of a given number are those numbers that can be divided by that given number without a remainder.

In other words, they are the products you get when you multiply that number with other numbers.

You can find the multiples of a number by multiplying it by another number, such as 2 x 1 = 2 and 2 x 2 = 4. Thus, 2 and 4 are multiples of 2. Incidentally, if the sum of the digits of a number is divisible by 3, then the number is a multiple of 3 (ex. 561→5 + 6 + 1 = 12).

Any number that is a multiple of all the given numbers is called a common multiple. The least common multiple is the smallest multiple of two or more numbers, such as how 24 is the least common multiple of 8 and 12.

Strategies for Finding the Greatest Common Factor and Least Common Multiple

Often, you are asked for either the Greatest Common Factor (GCF) or Least Common Multiple (LCM) of two numbers.

The best way to approach these types of problems is to look at each number’s prime factorization and make some decisions from that point. This is much more efficient than listing out all of the individual factors or multiples for each number! Look at the example below where we compare 24 and 30 to determine both the GCF and the LCM.

Using these tools for prime factorization, it becomes much more efficient to find the GCF and LCM of more than two numbers or for much larger numbers.

It is easy enough to find the GCF and LCM for numbers like 4 and 12, but when asked what is the GCF and LCM of 153 and 27, you’ll want to be using a prime factorization technique!

Practice Problems

1. Find all the factors of 30.
2. Find the common factors of 42 and 56. What is the greatest common factor (GCF)?
3. Find the prime factors of 24.
4. Find the first five multiples of 13.
5. Find the first three common multiples of 6 and 9. What is the least common multiple (LCM)?

Answers to Practice Problems

1. 1, 2, 3, 5, 6, 10, 15, 30
2. 1, 2, 7, 14 and the GCF is 14
3. 2, 3
4. 13, 26, 39, 52, 65
5. 18, 36, 54 and the LCM is 18