If an integer is divisible by $4$ and the sum of its last two digits is $13$, then what is the product of its last two digits?
I know that a number is divisible by $4$ if and only if its last two digits are divisible by $4$. So I need to find two digits that are divisible by $4$ and add up to $13$. I can list the pairs of digits that are divisible by $4$: $00, 04, 08, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96$. I can also list the pairs of digits that add up to $13$: $49, 58, 67, 76, 85, 94$. The only pair that appears in both lists is $76$. Therefore, the integer must end with $76$. The product of the last two digits is $7 \times 6 = 42$. 