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3

Part of 2015 South Africa National Olympiad

Problems(1)

Special divisors of $n$

Source:

8/8/2015
We call a divisor dd of a positive integer nn special if d+1d + 1 is also a divisor of nn. Prove: at most half the positive divisors of a positive integer can be special. Determine all positive integers for which exactly half the positive divisors are special.
number theory