Question 163
For a finite set $A$ of integers, denote by $|A|$ the cardinality of $A$, and denote by $\Sigma(A)$ the sum of all elements of $A$. Let $p$ be a prime number, and $A$ be a given set of positive integers such that $|A| \gt p$.
Let $N_{\text{even}}$ denote the number of subsets $B$ of $A$ such that $|B|$ is even and $\Sigma(B)$ is divisible by $p$. Let $N_{\text{odd}}$ denote the number of subsets $B$ of $A$ such that $|B|$ is odd and $\Sigma(B)$ is divisible by $p$. Then prove that $N_{\text{odd}} - N_{\text{even}}$ is divisible by $p$.
Let $N_{\text{even}}$ denote the number of subsets $B$ of $A$ such that $|B|$ is even and $\Sigma(B)$ is divisible by $p$. Let $N_{\text{odd}}$ denote the number of subsets $B$ of $A$ such that $|B|$ is odd and $\Sigma(B)$ is divisible by $p$. Then prove that $N_{\text{odd}} - N_{\text{even}}$ is divisible by $p$.