Question 34
A positive integer $N$ is called balanced, if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$.
a. Prove that there exist distinct positive integers $a$ and $b$ such that all the numbers $P(1), P(2), \ldots, P(50)$ are balanced.
b. Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.
a. Prove that there exist distinct positive integers $a$ and $b$ such that all the numbers $P(1), P(2), \ldots, P(50)$ are balanced.
b. Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.