Modern Cryptography and Elliptic Curves: Precursor to the Chinese Remainder Theorem

Exercise from P.62 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide.

Find the smallest number of marbles in a jar so that one remains if taken out 2, 3, 5 at a time, but none remain if taken out 11 at a time.

The answer obviously is a multiple of 11. And the answer obviously isn’t even. By writing out 33, 55, 77, 99, 121, \cdots, it’s easy to see that 121 is the first to satisfy such constraints.