<s>[INST] Induction proof: $2^n + 3^n ≡ 5^n (mod 6)$ I'm trying to prove that $2^n + 3^n ≡ 5^n\ (mod\ 6)$ using induction.
$n=1$:
$2+3≡5\ (mod\ 6)$
$n=k$:
$2^k + 3^k ≡ 5^k\ (mod\ 6)$
$n=k+1$:
$2^{k+1} + 3^{k+1} ≡ 5^{k+1}\ (mod\ 6)$
$2*2^k + 3*3^k ≡ 5*5^k\ (mod\ 6)$
$6\ |\ 5*5^k - 2*2^k - 3*3^k$
$6\ |\ (2+3)*5^k - 2*2^k - 3*3^k$
$6\ |\ 2*5^k + 3*5^k - 2*2^k - 3*3^k$
$6\ |\ 2*(5^k - 2^k) + 3*(5^k - 3^k)$
Not quite sure if I'm going in the right direction or where to go from here...
 [/INST] Multiply both sides of $$a^k+b^k\equiv(a+b)^k\pmod{ab},$$ by $a+b$ to find $$(a+b)^{k+1}\equiv(a+b)(a^k+b^k)\equiv a^{k+1}+b^{k+1}+a\cdot b^k+b\cdot a^k\pmod{ab}$$
Use the face $a\cdot b^k,b\cdot a^k,$ are both divisible by $ab$ for $k\ge1$
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