Counting 1 as prime would break the unique factorization that makes primes useful
Multiplying by 1 changes nothing, so if 1 were prime you could write 6 as 2 × 3, or 1 × 2 × 3, or 1 × 1 × 2 × 3, and so on forever. Every number would have endless factorizations instead of one. Mathematicians exclude 1 by definition so that each number keeps exactly one prime recipe.
