Question

What is the remainder when 6^17 17^6 is divided by 7?

Answer 1

62≡36≡1mod7

⟹616≡(62)8≡18≡1mod7

⟹617≡616∗6≡1∗6≡6mod7

⟹617≡6mod7

⟹17≡3mod7

⟹172≡3∗3≡2mod7

⟹174≡172∗172≡2∗2≡4mod7

⟹176≡174∗172≡4∗2≡1mod7

176≡1mod7

Now,  (617+176)≡(6+1)≡0mod7

Thus the remainder when (617+176) is divided by 7 is 0.

(617+176)=(7−1)17+(14+3)6≡(−1)17+36≡−1+729≡728≡0mod7

⟹(617+176)≡0mod7

Thus the remainder when (617+176) is divided by 7 is 0.