MathLabs

Problem 3

Prove that ∑k=0n(2n+12k+1)23k\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k} is not divisible by 55 for any integer n≥0n\ge0.
Step 5 of 5: Conclude
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2nS=α≠0(mod5)  ⟹  S≢0(mod5)2^nS=\alpha\ne0\pmod5\implies S\not\equiv0\pmod5
Detailed analysis

Since 2n2^n is invertible modulo 55 and 2nS=α≠02^nS=\alpha\ne0, the original sum SS is nonzero modulo 55. Therefore it is not divisible by 55 for every n≥0n\ge0.