M20-08

Official Solution:


If \(x\) and \(y\) are positive integers, is \(10^x - y\) divisible by 9 ?

Note that for any positive integer \(x\), \(10^x\) will have a digit sum equal to 1, so it will not be divisible by 3.

(1) \(y\) is divisible by 3.

In this case, \(10^x - y\) = (something not divisible by 3) - (something divisible by 3) = (something not divisible by 3). Since \(10^x - y\) is not divisible by 3, it is also not divisible by 9. Sufficient.

(2) \(10^x + y\) is not divisible by 9.

If \(y = 1\), the digit sum of \(10^x + y\) will be 2, so it won't be divisible by 9, but \(10^x - y\) will have only 9s as its digits, so it will be divisible by 9. However, if \(y = 2\), the digit sum of \(10^x + y\) will be 3, so it won't be divisible by 9, but \(10^x - y\) will have some number of 9s followed by an 8, so it won't be divisible by 9 either. Not sufficient.


Answer: A