Question

The storekeeper found that if he removed the apples from his box either 2,3,4,5 or 6 at a time there was always one left over. However, if he removed the apples 7 at a time there was no apples left. If the box holds 500 apples, how many apples does the storekeeper have?

Answer 1

Since the LCM of 2,3,4,5,6 is 60, there are 60n+1 apples in the box.

That number must also be a multiple of 7, so since 60 = 7*8+4, 5*60+1 = 5*7*8+5*4+1 = 7*5*8+21 = 7(5*8+3)

So, there are 301 apples in the box.

Answer 2

If you are working on number theory, look up Chinese remainder theorem.

If you have a puzzle or problem to solve, you can do it as follows:

If we took one apple from the box, then the number of apples must be divisible by 2, 3, 4, 5, and 6. Therefore the number of apples would be 60 or its multiples. (LCM of 2,3,4,5,6)

We look for a number (60 or its multiples) so that if we divide this number by 7, we should have 6 left, since we took one apple out earlier.

Try
60/7=8 with remainder 4 (or 8R4).
120/7=17R1, 180=25R5, 240=34R2, 300=42R6 (Good).

If we now throw back our apple to make 301, then it would satisfy all the given conditions.