2020 CIME I Problems/Problem 10
Problem 10
Let be the divisors of a positive integer . Let be the sum of all positive integers satisfying Find the remainder when is divided by .
Solution
Note that is even, as if is odd then the RHS is even and the LHS is odd. This implies that the first two divisors of are and . Now, there are three cases (note that represents a prime):
When . Then, is always odd, so this is a contradiction.
When . This implies that or . Then, .
When . If then . If then . This means as , however, this fails.
So, the sum of all possible values of are , so the remainder is . ~rocketsri
See also
2020 CIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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