(adsbygoogle = window.adsbygoogle || []).push({}); Let [tex]p_{1}, p_{2},...,p_{n}[/tex] be primes. Show that [tex]p_{1} p_{2}...p_{n}+1[/tex] is divisible by none of these primes.

Let [tex]p_{1}, p_{2},...,p_{n}[/tex] be primes

Let [tex]k \in N[/tex]

Assume [tex]p_{1}p_{2}...p_{n}+1=kp_{n}[/tex]

[tex]\frac{p_{1}p_{2}...p_{n}}{p_{n}}+\frac{1}{p_{n}}=k[/tex]

[tex]p_{1}p_{2}...p_{n-1}+\frac{1}{p_{n}}=k[/tex]

This is a contradiction because the left side will not be a natural number.

My issue is that this seems to only prove [tex]p_{1} p_{2}...p_{n}+1[/tex] is not divisible by [tex]p_{n}[/tex] and not all [tex]p_{1}, p_{2},...,p_{n}[/tex].

Thanks!

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# Homework Help: Prime Number Proof Help.

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