Step-by-step explanation:
For n=1, 19^n -13 = 19 -13 = 6, a multiple of 6.
For n = n+1, ...
19^(n+1) -13 = 19(19^n -13) +18·13
the factor (19^n -13) is divisible by 6 (for n=1), and ...
the factor 18·13 = 6·39 is divisible by 6.
Hence 19^n -13 is divisible by 6 for all n ≥ 1.
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Another way to look at this is to do modulo 6 arithmetic on the expression:
19^n -13 mod 6 = (19 mod 6)^n - (13 mod 6) = 1^n - 1 = 0
If the expression mod 6 is 0, then the expression is divisible by 6.