<s>[INST] Mathematical Induction Divisibility Problem Prove that if $n \ge 1$ is a positive integer, then $13^n − 6^n$ is divisible by $7$.
In proving the $n = k+1$ case, I get to $133k + 6^k\cdot13 - 6\cdot13^k = 7M$, where $M$ is a positive integer. $133k$ is divisible by $7$. How do I show the $6^k\cdot13 - 6\cdot13^k$ part is divisible by $7$?
 [/INST] $13^{k+1} - 6^{k+1} = 13^k* 13 - 6^k * 6 = 13^k(7 + 6) - 6^k*6 = 7*13^k + 6*13^k - 6^k*6 = 7*13^k + 6(13^k - 6^k) = 7*13^k + 6(7*M) = 7*(13^k + k)$ is a multiple of 7.
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