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

8

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 .