What are all the factors of 72?
The factors are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.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. Ill put the explanation in black in [brackets] and the answer in $color(blue)blue$.
Ill 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 thats 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 weve checked all the numbers up to and including 8. So were 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.]