Multiples, Factors and Prime Factors

Multiples are defined as the result of multiplying a number by an integer, while factors are numbers that divide another number without leaving a remainder. Prime factors, specifically, are the prime numbers that can be multiplied together to yield the original number.

The MULTIPLES of a number are just its times table.

EXAMPLE
Find the first 8 multiples of 13.

  1. 13

  2. 26

  3. 39

  4. 52

  5. 65

  6. 78

  7. 91

  8. 104

The factors of a number are all the numbers that divide into it.

There’s a method that guarantees you’ll find them all:

1) Start off with 1 x the number itself, then try 2 x, then 3 x and so on, listing the pairs in rows.

2) Try each one in turn. Cross out the row if it doesn’t divide exactly.

3) Eventually, when you get a number repeated, stop.

4) The numbers in the rows you haven’t crossed out make up the list of factors.

EXAMPLE

Find all the factors of 24.

Start by dividing 24 by the smallest prime numbers, which are 1, 2, 3, and so on, until you reach the number itself. This will give you the pairs of numbers that multiply together to equal 24, which are: 1 and 24, 2 and 12, 3 and 8, and 4 and 6. Thus, the complete list of factors of 24 is: 1, 2, 3, 4, 6, 8, 12, and 24.

Any number can be broken down into its prime factors by continuing to divide by prime numbers until only prime numbers remain. For 24, the prime factorization can be represented as: 2 x 2 x 2 x 3, or in exponential form as 2^3 x 3^1.

To write a number as a product of its prime factors, use the Factor Tree method:

  1. Start with the number 24 at the top of the tree.

  2. Divide it by the smallest prime number, which is 2, to get 12.

  3. Continue dividing 12 by 2 to get 6, and then again divide 6 by 2 to reach 3.

  4. Since 3 is a prime number, stop here.

  5. The branches of the tree will show that 24 can be broken down into the factors 2, 2, 2, and 3, confirming that the prime factorization is indeed 2^3 x 3^1.

EXAMPLE
Express 420 as a product of prime factors.

So 420 = 2 × 2 × 3 × 5 × 7 = 2^2 × 3^1 × 5^1 × 7^1. This process demonstrates how any composite number can be expressed as a unique product of prime factors, which is fundamental in number theory.

No matter how large or small the number, the principle of prime factorization remains consistent, proving essential for calculations involving multiples, factors, and divisibility.

Follow the methods above to find factors and prime factors

Make sure you know the Factor Tree method inside out, then give these Exam Practice Questions a go…

Q1) Use the following list of numbers to answer the questions below. 4, 6, 10, 14, 15, 17, 24, 30

a) Find one number that’s a multiple of 2, a multiple of 3 and a multiple of 4. One such number is 24, as it can be divided evenly by 2, 3, and 4.

b) Find one number that’s a multiple of 3 and a factor of 36. One such number is 12, since it is divisible by 3 and can be used to evenly divide 36.

Q2) What number should replace the to make 14 × 30 = 7 x _ 14 true? The missing number is 60, as multiplying 7 by 60 and then by 14 yields the same product as 14 times 30.

Q3) Express 990 as a product of its prime factors. To express 990 as a product of its prime factors, we start by dividing 990 by the smallest prime number, which is 2, giving us 495. Next, we divide 495 by 3, resulting in 165. Continuing this process, we divide 165 by 3 again to get 55 and then divide 55 by 5, yielding 11, which is a prime number. Hence, the prime factorization of 990 is 2 × 3^2 × 5 × 11.