<P> Assume that s> 1 is the smallest positive integer which is the product of prime numbers in two different ways . If s were prime then it would factor uniquely as itself, so there must be at least two primes in each factorization of s: </P> <Dl> <Dd> s = p 1 p 2 ⋯ p m = q 1 q 2 ⋯ q n . (\ displaystyle (\ begin (aligned) s& = p_ (1) p_ (2) \ cdots p_ (m) \ \ & = q_ (1) q_ (2) \ cdots q_ (n). \ end (aligned))) </Dd> </Dl> <Dd> s = p 1 p 2 ⋯ p m = q 1 q 2 ⋯ q n . (\ displaystyle (\ begin (aligned) s& = p_ (1) p_ (2) \ cdots p_ (m) \ \ & = q_ (1) q_ (2) \ cdots q_ (n). \ end (aligned))) </Dd> <P> If any p = q then, by cancellation, s / p = s / q would be another positive integer, different from s, which is greater than 1 and also has two distinct factorizations . But s / p is smaller than s, meaning s would not actually be the smallest such integer . Therefore every p must be distinct from every q . </P>

The fundamental theorem of arithmetic is applicable to