Question 108
For a positive integer $m$ denote $S(m)$ the sum of its natural divisors, and if $n$ and $p$ are positive integers, denote $Q(n, p)$ the sum of the quotients of the division of $n$ by the natural divisors of $p$ (for instance, $Q(18, 10) = 18 + 9 + 3 + 1 = 31$).
Let $a$ and $b$ be two positive integers.
a) Prove that, if $S(a) = Q(a, b)$ and $S(b) = Q(b, a)$, then $a = b$.
b) Is it always true that, if $S(a) + S(b) = Q(a, b) + Q(b, a)$, then $a = b$?
Let $a$ and $b$ be two positive integers.
a) Prove that, if $S(a) = Q(a, b)$ and $S(b) = Q(b, a)$, then $a = b$.
b) Is it always true that, if $S(a) + S(b) = Q(a, b) + Q(b, a)$, then $a = b$?