Answer = 0

Imp to note that Mod is a linear function

(1^7 + 2^7 + 3^7 + 4^7 + ...+100^7 )mod4

= (1^7+2^7+3^7+4^7)mod4 + (5^7+6^7+7^7+8^7+...+100^7)mod4

Observe that

(5^7+6^7+7^7+8^7+...100^7)mod4 = 5^7mod4 + 6^7mod4 + 7^7mod4 +8^7mod4 +...+ 100^7mod4

Now,

5^7mod4 = 1^7mod4

6^7mod4 = 2^7mod4

7^7mod4 = 3^7mod4

8^7mod4 = 4^7mod4 and so on the pattern will repeat

Ultimately, we have

25*(1^7 +2^7+3^7+4^7)mod4

=1*(1 + 0 + 3 + 0) mod4

=1*4mod4

=0

{ Imp to note here is that 1^7 = 1 so that was easy; 2^7/2^2 = 2^5 so that tells us remainder 0;

Analyse 3^1mod4 = 3; 3^2mod4 = 1; 3^3mod4 = 3; 3^4mod4=1; 3^5mod4=3; 3^6mod4=1 and 3^7mod4 = 3; this is a helpful observation if the number 7 was higher; and lastly 4^7mod4 is defintely 0.}