 To see this, assume the contrary, that there are two independent factorizations of "L" into "m" and "n" prime factors, respectively

Since each prime "p" divides "L" by assumption, it must also divide one of the "q" factors; since each "q" is prime as well, it must be that "p" = "q".Iteratively dividing by the "p" factors shows that each "p" has an equal counterpart "q"; the two prime factorizations are identical except for their order.The unique factorization of numbers into primes has many applications in mathematical proofs, as shown below.