Question 176
We call a natural number $n$ balanced when $n=1$ applies or if $n$ can be written as a product of an even number of (not necessarily different) prime factors. For every pair of $(a, b)$ positive integers, $P(x) = (x+a)(x+b)$.
a) Are there two different positive integers $a$ and $b$ for which all numbers $P(1), P(2), \ldots, P(50)$ are balanced?
b) Man Proof: If $P(m)$ is balanced for all positive integers $m$, then $a = b$ applies.
a) Are there two different positive integers $a$ and $b$ for which all numbers $P(1), P(2), \ldots, P(50)$ are balanced?
b) Man Proof: If $P(m)$ is balanced for all positive integers $m$, then $a = b$ applies.