Thanks guys for posting. For the first one it is 576 and Jeff's way makes sense, but I found another way and I don't know if this for sure works.

1) since (9,16)=1 then the greatest common multiple of the 2 is 144. I then found all the factors of 144, which are 1,2,3,4,8,9,12,16,18,36,48,72,144 which 13 numbers so this cant be it. It has to be a multiple of 144 though.

Next is 288, now I just took all the factors of 144 and multiplied them by 2 and added them to the list because these will also be factors of 288, this gives us 6,24,32,96,288 also as divisors which makes the total number of divisors to 18, so this still isn't it.

Next is 432, I took the original 13 factors of 144 and multiplied them by 3 then added them to the list. This gives us 6,24,27,54,108,216,432 also as divisors which makes the total number to 20, so this still doesn't work.

Lastly is 576 which is the answer. For this one I took the 18 factors of 288 and multiplied them by 2 because 4 isn't prime so I wouldn't get all the factors if I did 4 times the factors of 144, i.e. 6 would be left out and that is a factor of 576. This gives us 62,192,576 on top of the 18 factors of 288, that is 21 divisors of 576 with 9 and 16 being divisors so this is the answer.

For the second one I just found out that they are just positive integers so the largest is 70 with the other 3 being 1,2,3

If you have another method for the first one that would be great so my mom can explain it to her class using different ways.