= Nonunique factorization in a congruence submonoid
{title2=$21\cdot221=51\cdot91$}
The <congruence submonoid of the positive integers> with modulus ten has four distinct <irreducible elements of a commutative monoid> $21,221,51,91$ with equal pairwise products. Their ordinary prime factors are respectively $(3,7),(13,17),(3,17),(7,13)$. None of these proper prime factors is congruent to one modulo ten. The <Euclid lemma> proves that a product of two distinct primes has no other proper positive divisors, so all four are irreducible in this monoid. This gives nonunique factorization without assuming the fundamental theorem of arithmetic.
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