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Let $n$ and $d$ denote integers. We say that $d$ is a divisor of $n$ if $n = cd$ for some integer $c$. An integer $n$ is called a prime if $n > 1$ and if the only positive divisors of $n$ are $1$ and $n$. Prove, by induction, that every integer $n > 1$ is either a prime or a product of primes.
My try: First, that there's nothing to prove because a number is always a prime or not, so do not what to think. Step: $P(n): n$ is either a prime or a product of primes. If n=2 then 2 is prime. $P(n): True$ I want to see $P(n) \rightarrow P(n+1)$ If $n$ is a prime then $2$ is a divisor of $p+1$, then is a product of primes. If $n$ is a product of primes... I can't say anything about $n+1$. Some help... please.