2021 JMC 10 Problems/Problem 19
Problem
Two distinct divisors of are mutual if their difference divides their product. For instance,
is mutual as
Suppose a mutual pair
exists where
for a positive integer
What is the sum of all possible
Solution
Observe that Because
, it follows that
. Since
, we must also have
. Note that we cannot have
, because this will result in
being neither a multiple of
nor
. Thus, we need only to check
and
Note that
must be of the form
for non-negative integers
and
so our desired answer is
.