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