When an integer is divided by 15, the remainder is 7. Find the sum of the remainders when the same integer is divided by 3 and by 5.
One way to do that is to add 7 to any multiple of 15. For example, 15 + 7 = 22, or 30 + 7 = 37, or 45 + 7 = 52. Any of these numbers will have a remainder of 7 when divided by 15. Now I want to find the remainders when these numbers are divided by 3 and by 5. For 22, the remainder when divided by 3 is 1, and the remainder when divided by 5 is 2. For 37, the remainder when divided by 3 is 1, and the remainder when divided by 5 is 2. For 52, the remainder when divided by 3 is 1, and the remainder when divided by 5 is 2. I notice a pattern here: no matter what multiple of 15 I add 7 to, the remainders when divided by 3 and by 5 are always 1 and 2, respectively. So the sum of the remainders when the same integer is divided by 3 and by 5 is always 1 + 2 = 3. 