Question 136
Given a positive integer $n$, we denote by $n!$ (‘n factorial’) the number we get if we multiply all integers from $1$ to $n$. For example: $5! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120$.
a. Determine all integers $n$ with $1 \le n \le 100$ for which $n! \cdot (n+1)!$ is a perfect square. Also, prove that you have found all solutions $n$.
b. Prove that no positive integer $n$ exists such that $n! \cdot (n+1)! \cdot (n+2)! \cdot (n+3)!$ is a perfect square.
a. Determine all integers $n$ with $1 \le n \le 100$ for which $n! \cdot (n+1)!$ is a perfect square. Also, prove that you have found all solutions $n$.
b. Prove that no positive integer $n$ exists such that $n! \cdot (n+1)! \cdot (n+2)! \cdot (n+3)!$ is a perfect square.