A question defines A = {1,2,3,4,5,6} and a binary operation * such that $a*b=r$, where r is the least non-negative remainder when the product $ab$ is divided by $k$. Find k for * to be a binary operation.
The answer is five, apparently, but;
1: What's up with the "non-negative remainder" thing? Are remainders ever negative?
2: On trying out 1 and 5 as $a$ and $b$, the remainder on dividing it by k(which is 5) is 0. And if 0 is acceptable as th least possible remainder, then k wouldn't be 5, it would be 1, and the remainder in each case would be zero, wouldn't it?
Why did it work out this way?