jmc

Question

The number of students in Teresa's graduating class is more than 50 and fewer than 100 and is 1 less than a multiple of 3, 2 less than a multiple of 4, and 3 less than a multiple of 5. How many students are in Teresa's graduating class?

Accepted Answer

Let the number of students in Teresa's class be a. Then align* a -1 2 3, a -2 2 4, a -3 2 5. align* Since (3,4)=(4,5)=(3,5)=1, we have a 23 4 5, that is, a 260. Since a is then of the form a=2+60n, the only such number in the range 50<a<100 is 62.