Exercise 1837Number theory › PrimesOlympiadAdded 3 Nov 2026

Primorial and GCD

#primorial#gcd#primes

Statement

Let $p_n$ be the $n$ -th prime and $\#_n=p_1\cdots p_n$ the primorial. Show that $\#_n+1$ is coprime to every prime $\le p_n$ .

Hint

Any prime $p\le p_n$ divides $\#_n$ ; if it also divides $\#_n+1$ , then it divides $1$ .

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Setup. $p_i\mid\#_n$ .

Contradiction. If $p_i\mid\#_n+1$ , then $p_i\mid(\#_n+1)-\#_n=1$ , impossible.

Conclusion. $\gcd(\#_n+1,p_i)=1$ for all $i\le n$ ; so any prime divisor of $\#_n+1$ is $>p_n$ .