Problem:
Find the sum of all positive integers , where and are non-negative integers, for which is not a divisor of .
Solution:
Note that
is not an integer if and only if or , that is, if and only if
When both and , there are no values of and that satisfy . When reduces to , which is satisfied only by . When , ( ) reduces to , which is satisfied only by . Thus there are six values of for which not a divisor of , namely, , and , and their sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions