Question

Giúp mình. CMR : \(3^{2^{4x+2}}\) + 2^{\(3^{4x+1}\)} +5 \(⋮\) 22

Answer 1

Dễ thấy \(3^{2^{4x+2}}+2^{3^{4x+1}}+5⋮2\left(1\right)\)

Ta chứng minh nó chia hết cho 11.

Ta có: \(2^{4x+1}=4.16^x\equiv2\left(mod5\right)\)

\(\Rightarrow3^{2^{4x+1}}=3^{5m+2}=9.243^m\equiv9\left(mod11\right)\)

Ta có: \(3^{4x+1}=3.81^x\equiv3\left(mod10\right)\)

\(\Rightarrow2^{3^{4x+1}}=2^{10n+3}=8.1024^n\equiv8\left(mod11\right)\)

\(\Rightarrow3^{2^{4x+1}}+2^{3^{4x+1}}+5\equiv9+8+5\equiv22\equiv0\left(mod11\right)\)

\(\Rightarrow3^{2^{4x+1}}+2^{3^{4x+1}}+5⋮11\left(2\right)\)

Từ (1) và (2) \(\Rightarrow3^{2^{4x+1}}+2^{3^{4x+1}}+5⋮22\)

PS: Sửa đề luôn rồi nhé