What are all the numbers that when you times them together you get 72?

Answer 1

Factoring is crucial to algebra and any math built on algebra. Like any other skill you get better at this one with practice, but it helps to have a method. I like to do both ends against the middle. By that I mean, if you think of 72 as a number line, what you want to do is find the factors at the ends and work you way to the middle of the line, that way you only have to test the smaller half of the number line. Having an orderly system means that you will be sure not to miss any and get there faster. You can always just try them all but for larger numbers that would be time consuming. Here goes...

All numbers are divisible by 1 and by themselves.

1 x 72 = 72

So, we now have 1, 72

72 is an even number so it is also divisible by 2.

2 x 36 = 72

So, now we have 1, 2, 36, 72

Any other numbers we find will be between 2 and 36, less that half of the total numbers!

I know that 72 is divisible by 3. If you add the digits of a number together and get a multiple of 3, the original number is also a multiple of 3! 7 + 2 = 9. Yep. (Special bonus, if they add up to 9, the original number is also divisible by 9.)

3 x 24 = 72

Now we have 1,2,3,24,36,72

Now, this number is also divisible by 4. You can see that some of the factors are divisible by 4.

4 x 18 = 72

Now it's 1,2,3,4,18,24,36,72

We can rule out 5. It doesn't end in 0 or 5.

It is definitely divisible by 6. 2 and 3 are factors and they make 6.

6 x 12 = 72

We are getting very close to the middle now. 6 and 12 are pretty close.

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

7 won't work. 7 goes into 70 (obviously) and cannot go into 72.

8 is easy. We have 2 and 4 so we have to have 8.

8 x 9 = 72

Hey, those meet! That means we are done. There aren't any other numbers left to test.

The final answer is 1,2,3,4,6,8,9,12,18,24,36,72.

Frequently, you just need the "prime factors." That means that if you list 2 or 3 you don't need to list 4 or 9 as well, or if you list 2 and 3 you don't need to list 12. Prim numbers only have two factors, 1 and itself.

So, 2 is prime (it is the only even prime). It factors to 1 and 2.

But, 4 is not prime. It factors to 1, 2, 4.

The prime factors of 72 are 1,2,3. That's it. All the other factors are just multiples of 2 and 3.

Later in math you will learn a faster way to do this using the quadratic formula so don't worry about having to do this forever.

Hope this helps!

D