 Nonunique factorization in a congruence submonoid

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.