courtesy of a student

If (x # y) represents the remainder that results when the positive integer x is divided by the positive integer y, what is the sum of all the possible values of y such that (16 # y) = 1?

8
9
16
23
24

like all other problems featuring strange symbols, this one is fundamentally based upon the idea of following directions. all the information you need to solve the problem is buried in the definition of the symbol; all you have to do is obey the content of the definition, and perhaps manipulate it in clever ways.

in this problem, we can use the given definition to rephrase the question as follows:
what is the sum of all positive integers y such that 16 divided by y gives a remainder of 1?

this may seem like an almost unlimited set of possibilities at first, but you'll soon realize that all of the y values that work must be less than 16. (reason: if you divide by a number larger than 16, then 16 itself will be the remainder.)

at that point you can either just do trial and error, or you can make the following realization: if dividing 16 by some number leaves a remainder of 1, then that number must go into 15. (think about what a remainder means - it's 'leftovers' after you do the division - and this should make sense.)
either way, you'll find out that the numbers that work are 3, 5, and 15, so the sum is 23. (note that 1 doesn't work, as division by 1 always leaves a remainder of 0.)