<s>[INST] Finding the remainder when $2^{100}+3^{100}+4^{100}+5^{100}$ is divided by $7$ 
Find the remainder when $2^{100}+3^{100}+4^{100}+5^{100}$ is divided by $7$.

Please brief about the concept behind this to solve such problems. Thanks.
 [/INST] 2^100 + 3^100 + 4^100 + 5^100/7 
= (2^3)^33 *2 + (3^3)^33 *3 + (4^3)^33 *3 + (5^3)^33 *3 / 7 
= 2*8^33 + 3*27^33 + 4*64^33 + 5*125^33 / 7
= 2*1^33 + 3*-1^33 + 4*1^33 + 5*-1^33 / 7
= 2*1 + 3*-1 + 4*1 + 5*-1 / 7
= 2 - 3 + 4 -5 / 7 
= -2/7
= 7-2
= 5 remainder
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