Question

Prove that a(a + 1)(2a + 1) is divisible by 6 for integer a using a quicker proof of this, based on the observation that 6 I m (6 divides m) if and only if 2 I m (2 divides m) and 3 I m (3 divides m).

Please use modulo congruences.    

Answer 1

is even. Solution. The product ninti) Int2) E 01 mod 2) Since if his even the product will be even, & if he is odd then (M+1) will be even & again the product We also know that ninti (n+2) = 0 (mod 3) le can prove this by notating that in mod 3 me must be 0,1, 082 & all of these cases imply the product is divinble by 3. Case Ir ne or mods) ninu (n+2) = o[med 3) Case Inn Elcmod 3) >> n+2 = 0 (mod 3) => n(n+n(n+2) = 0 mod 3. Case Ihr n=2(mod 3) = n+1 = o(mod 3) -> ninos (+2) = 0 [mod 3) Therefore ninth in +2) h divisible by 2 & 3 which implecs it is divisible by 'b.