2021 JMC 10 Problems/Problem 19
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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 ![\[(d_1-d_2) |d_1 d_2 \implies (k-1)d_2 | k\cdot (d_2)^2 \implies (k-1) | k d_2.\]](http://latex.artofproblemsolving.com/9/d/f/9df32205c56f8968c903f1f84815a4a4db118bc1.png)
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 
.
