Question

What are all the factors of 72?

Answer 1

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

Explanation:

I find factors in pairs, It will look like more work than it is, because I will explain how I am doing these steps. I do most of the work without writing it down. I'll put the explanation in black in [brackets] and the answer in `color(blue)"blue"`.

I'll proceed by starting with `1` on the left and checking each number in order until either I get to a number already on the right or I get to a number greater than the square root of 72.

`color(blue)(1 xx 72)`

[I see that 72 is divisible by 2, and do the division to get the next pair]

`color(blue)(2 xx 36)`

[Now we check 3 and we get the next pair.]
[I use a little trick for this. I know that 36 is divisible by 3 and `36 = 3xx12`. This tells me that `72 = 2xx3xx12`, so I know that `72 = 3xx2xx12 = 3xx24`]

`color(blue)(3 xx 24)`

[Now we need to check 4. Up above, we got `72 = 2xx36` since `36 = 2xx18`, we see that `72 = 2xx2xx18 = 4xx18`]

`color(blue)(4 xx 18)`

[The next number to check is 5. But 72 is not divisible by 5. I usually write a number before I check, so if a number is not a factor, I cross it out.]

`color(blue)cancel(5)`

{Move on to 6. Looking above I want to 'build' a 6 by multiplying a number on the left times a factor of the number to its right. I see two ways to do that: `2xx36 = 2xx3xx12 = 6xx12` and `3xx24 = 3xx2xx12=6xx12`. (Or maybe you just know that `6xx12=72`.)]

`color(blue)(6 xx 12)`

[72 is not divisible by 7.]

`color(blue)cancel(7)`

{`4xx18 = 4xx2xx9=8xx9`]

`color(blue)(8 xx 9)`

[And that's all. 9 and the factors that are greater than 9 are already written on the right in the list of pairs above.]
[Is that clear? Any factor of 72 greater than 9 must be multiplied by something less than 8 to get 72. But we've checked all the numbers up to and including 8. So we're finished.]

[If we were doing this for `39` we would get `1xx39` and `3xx13`, then we cross off every number until we notice that `7xx7 = 49`. If 39 had a factor greater than 7 it would have to be multiplied by something less that 7 (otherwise we get 49 or more). So we would be finished.]