400 (number) sum of twelve consecutive primes (13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59) 413 = 7 × 59, Mertens function returns 0, self number 414 = 2 × 3 × 23, Mertens function returns 0, nontotient, Harshad number 415 = 5 × 83, 416 = 2 × 13 417 = 3 × 139 418 = 2 × 11 × 19, sphenic number, A prime number, Sophie Germain prime, Chen prime, Eisenstein prime with no imaginary part, highly cototient number, Mertens function returns 0 A prime
Prime number corresponds to "congruence" in modular arithmetic: formula_22 and formula_121 are congruent (written formula_122 mod formula_1) when they have the same remainder after division by formula_1. However, in this system of numbers, division by all nonzero numbers is possible if and only if the modulus is prime. For instance, with the prime number formula_125 as modulus, division by formula_28 is possible: formula_127, because clearing denominators by multiplying both sides by formula_28 gives the valid formula formula_129. However, with the composite modulus formula_130, division by formula_28 is impossible. There is no valid solution to formula_132: clearing denominators by multiplying by formula_28 causes