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The prime factorization of 72 is 2 x 2 x 2 x 3 x 3. That and the rules of divisibility tell us that 72 is divisible by 2, 3, 4, 6, 8 and 9. That should be enough information.

Write down the number and itself.

72 and 1 Divide 72 by 2

36 and 2 Divide 72 by 3

24 and 3 Divide 72 by 4

18 and 4 Divide 72 by 6

12 and 6 Divide 72 by 8

9 and 8

Answer:1,2,3,4,6,8,9,12,18,24,36,72

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Find the prime factorisation of 72 in power format.

Now multiply together all the primes for all the combinations of their powers from 0 to their maximum power which is the power of the factorisation. This is most easily done by:

- Write the primes all to the power 0
- Multiply the primes together to their relative powers (any prime to the power 0 is 1) to get the next factor
- Increase the power of the first prime by 1
- If this is not greater than its value in the prime factorisation, repeat from step 2.
- Reset the first prime's power to zero and increase the next prime's power by 1.
- If this is not greater than its value in its prime factorisation, repeat from step 2.
- Reset this prime to zero and increase the next prime's power by 1
- Keep repeating this until all the primes have gone through all their possible powers from 0 to their maximum value.

So for 72 this is:

72 = 23 x 32

⇒ its factors are:

20 x 30 = 1 x 1 = 1

21 x 30 = 2 x 1 = 2

22 x 30 = 4 x 1 = 4

23 x 30 = 8 x 1 = 8

20 x 31 = 1 x 3 = 3

21 x 31 = 2 x 3 = 6

22 x 31 = 4 x 3 = 12

23 x 31 = 8 x 3 = 24

20 x 32 = 1 x 9 = 9

21 x 32 = 2 x 9 = 18

22 x 32 = 4 x 9 = 36

23 x 32 = 8 x 9 = 72

So the factors of 72 are: 1, 2, 4, 8, 3, 6, 12, 24, 9, 18, 36, 72

Or in ascending order: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

By the rules of divisibility, you know that 72 is divisible by 1, 2, 3, 4, 6 and 8. Dividing those numbers into 72 gives you the other half.

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

Since 36 is a factor of 72, all of its factors are included in the factors of 72. Just add 72 to the list of factors of 36.

They are: 1 2 3 4 6 8 9 12 18 24 36 and 72

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Q: How do you use the prime factorization of 72 to find all of its factors?

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