Question

Prove that the integer 53^103 + 103^53 is divisible by 39 and 111^333 + 333^111is divisible by 7

Answer 1

As 39=13⋅339=13⋅3

For non-negative integers m,nm,n

53≡1(mod13)⟹53n≡153≡1(mod13)⟹53n≡1 and 103≡−1(mod13)⟹10353≡(−1)53103≡−1(mod13)⟹10353≡(−1)53

⟹53103+10353≡1+(−1)(mod13)⟹53103+10353≡1+(−1)(mod13)

and 53≡−1(mod3)⟹53103≡(−1)10353≡−1(mod3)⟹53103≡(−1)103 and 103≡1(mod3)⟹103m≡1103≡1(mod3)⟹103m≡1

⟹53103+10353≡−1+(1)(mod3)