The fundamental theorem of arithmetic has two important consequences. One is regarding the divisors of a number, the other is regarding gcd and lcm.
If \(n = p_1^{a_1}\cdots p_k^{a_k}\) is the prime factorization of the number \(n\), then any positive divisor \(d\) of \(n\) has the form:
\[d = p_1^{b_1}\cdots p_k^{b_k}\]
where each \(b_i\leq a_i\).
In particular, there are exactly \((a_1 + 1)(a_2 + 1)\cdots(a_k + 1)\) different positive divisors of \(n\).
Suppose \(n = p_1^{a_1}\cdots p_k^{a_k}\) and \(m = p_1^{b_1}\cdots p_k^{b_k}\) are prime number factorizations of \(n\) and \(m\), where we have allowed some of the exponents to equal \(0\) to ensure we have the same list of prime numbers. Then we have:
\[\gcd(n, m) = p_1^{\min(a_1, b_1)}p_2^{\min(a_2,b_2)}\cdots p_k^{\min(a_k, b_k)}\]
\[lcm(n, m) = p_1^{\max(a_1, b_1)}p_2^{\max(a_2,b_2)}\cdots p_k^{\max(a_k, b_k)}\]
A direct consequence of this is the formula \(nm = \gcd(n, m)lcm(n,m)\) that we saw earlier.