Show that 54 x 4^(6n) + 6^n x 11^(n + 1) is divisible by 65
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4^(6n) = (4^6)^n = (64^2)^n
= ((-1)^2)^n mod 65
= 1 mod 65
6^n x 11^(n+1)
= 11 x 66^n
= 11 x 1^n mod 65
= 11 mod 65
So in mod 65,
N = 54 x 1 + 11 mod 65
= 65 mod 65
= 0 mod 65
= ((-1)^2)^n mod 65
= 1 mod 65
6^n x 11^(n+1)
= 11 x 66^n
= 11 x 1^n mod 65
= 11 mod 65
So in mod 65,
N = 54 x 1 + 11 mod 65
= 65 mod 65
= 0 mod 65