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THE FUNDAMENTAL THEOREM OF ARITHMETIC Suppose that an integer n > 1 has two factorizations into irreducibles, n = p1 · · · pk = q1 · · · q` , all pi and all qj irreducible. Thus pk | q1 · · · q` , and so since the irreducible pk is prime, in fact pk | qj for some j ∈ {1, · · · , `}. Since multiplication commutes we may freely re-index the q’s so that pk | q` . But q` is irreducible, so its only factors are 1 and itself. Since pk is a factor of q` and pk 6= 1, we have pk = q` . So now we may cancel the rightmost terms of the equality p1 · · · pk = q1 · · · q` (where now we know pk = q` ) to get a new equality n0 = p1 · · · pk−1 = q1 · · · q`−1 , n0 < n. By Strong Induction, n0 has a unique factorization. That is, k − 1 = ` − 1 and pi = qi for i ∈ {1, · · · , k − 1}. Consequently the two factorizations of n agree as well. That is, the factorization of n is unique. 1