The number is a common factor of 96 and 144 the sum of the number's digits is 7. What is the number?

Answer 1

The easiest way to figure this out is probably to look at what the factors of 96 and 144 are, eliminate the ones that are not common, and then see which ones have a pair of digits that add up to seven.

So let's look at all of the factors of 96:
2, 3, 4, 6, 8, 12, 16, 24, 32, 48

And then all of the factors of 144:
2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72

Now we can intersect those two sets, keeping only the numbers they have in common:
2, 3, 4, 6, 8, 12, 16, 24, 48

We're told that the sum of the digits is 7, and seven is not a factor of either, so it must be a two digit number. With that in mind, we can reduce the set by eliminating all of the one-digit numbers:
12, 16, 24, 48

It's pretty obvious what the answer is at this point, but let's add one more logical step: we know that the sum of the digits is seven. That means that we can eliminate all numbers that have a digit greater than or equal to seven. That leaves us with a smaller set still:
12, 16, 24

And with this reduced set, it's pretty easy to see which value works:
1 + 2 = 3
1 + 6 = 7
2 + 4 = 6

So the answer is 16.