Smallest noncyclic quotients of braid and mapping class groups

We show that the smallest non-cyclic quotients of braid groups are symmetric groups, proving a conjecture Margalit. Moreover we recover results Artin and Lin about classification homomorphisms from on n strands to k letters, where is at most n. Unlike original proofs, our method does not use Bertrand-Chebyshev theorem, answering question Artin. Similarly for mapping class group closed orientable surfaces, quotient given by mod two reduction symplectic representation. provide an elementary proof this result, originally due Kielak-Pierro, which proves Zimmermann.