Question

Find the smallest positive integer that leaves a remainder of 1 when divided by 2, a remainder of 2 when divided by 3, a remainder of 3 when divided by 4, ... and a remainder of 7 when divided by 8

Answer 1

The answer is 11. When divided by 2 it gives 5, remainder 1; when divided by 3 it gives 3, remainder 2; and when divided by 4 it give 2, remainder 3.

Note that the answer must be an odd number, since when divided by 2 it leaves a remainder of 1. So 3, 5, 7, 9, 11, etc. However, 3 does not work because when divided by 3 it gives a remainder of 0, same for 9. 5 does not work because when divided by 4 it gives a remainder of 1. And 7 does not work because when divided by 3 it has remainder 1. So the 3, 5, 7, and 9 won’t work. The first number that will work is 11.