Exercise 922Number theory › Divisibility and gcdL1Added 3 Nov 2026

Semiprimes and divisors

#semiprime#divisors#factorization

Statement

Let $n=pq$ be the product of two distinct primes $p<q$ . Compute $\tau(n)$ and $\sigma(n)$ .

Hint

Use the formula for $\tau$ and $\sigma$ on prime factorizations.

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Number of divisors. $\tau(n)=(1+1)(1+1)=4$ : divisors $1,p,q,pq$ .

Sum of divisors. $\sigma(n)=(1+p)(1+q)=1+p+q+pq$ .

Conclusion. $\tau(pq)=4$ , $\sigma(pq)=(1+p)(1+q)$ .

Application. If $n$ has exactly $4$ divisors, $n=p^3$ or $n=pq$ with $p<q$ primes.