Find the remainder when 36728^(1^2+2^2+3^2+..........50^2) is divided by 15.

Hi (1^2+2^2+3^2+4^2+......... +50^2) = (50)(51)(101)/6 =42925 Now we know that 15 = 3*5 so using CRT we get 1. 36728^42925 mod 3 = (-1)^4295 mod 3 = -1 mod 3 = 2 Now taking with 5 36728^42925 mod 5 = 3^42925 mod 5 since 3 and 5 are co-prime we get 3^1 mod 5 ( a^b mod x = a^m mod x where m = b mod Euler (x) ) we get 3 Therefore we get : 3m+2=5k+3 so m=2 , k=1 we get remainder as 8 .

8