Exercise 1120Number theory › PrimesL1Added 3 Nov 2026

Factorial and coprime integers

#factorial#coprime#bounds

Statement

Let $n\ge1$ . Show that $n!$ and $n!+1$ are coprime.

Hint

Any common divisor divides $1$ .

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Difference. If $d\mid n!$ and $d\mid n!+1$ , then $d\mid1$ , so $d=1$ .

Conclusion. $\gcd(n!,n!+1)=1$ .

Generalization. $\gcd(n!,n!+k)=1$ for $1\le k\le n$ (since any prime divisor of $k$ divides $n!$ , so it cannot also divide $n!+k$ ).