How many houses did the two children count during Halloween?

Question

While trick or treating in Clayton, NY (population 1,978), some friends play a counting game-they counted all the houses they visited while trick or treating (they all counted the same houses). Diego counted the houses by 5's, and had four houses left at the end. Madeline counted the houses by 7's, and had six houses left over at the end. Based on Diego's and Madeline's counts, how many houses did they visit? Is this the only possible answer? Explain why you are sure you found all possible counts.

Answer 1

You can start by listing the multiples of each number (5 & 7) and compare.

5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60

+4: 9, 14, 19, 24, 29, 34, 39, 44, 49, 54, 59, 64

7, 14, 21, 28, 35, 42, 49, 56, 63, 70

+6: 13, 20, 27, 34, 48, 55, 62, 69, 76

Notice, that when you add the remainder to each corresponding amount, you can find where the numbers overlap. The first number they have in common is 34. So, one possible answer would be 34.