Question

cho n là số nguyên dương lẻ, CMR \(1+2^n+3^n+4^n+5^n\) chia hết cho 15

Answer 1

\(n=2k+1\)

\(\Rightarrow A=1+2.4^k+3.9^k+4.16^k+5.25^k\)

- Ta có: \(4\equiv1\left(mod3\right)\Rightarrow2.4^k\equiv2mod\left(3\right)\)

\(16\equiv1\left(mod3\right)\Rightarrow4.16^k\equiv1\left(mod3\right)\)

\(25\equiv1\left(mod3\right)\Rightarrow5.25^k\equiv2\left(mod3\right)\)

\(\Rightarrow A\equiv\left(1+2+1+2\right)\left(mod3\right)\Rightarrow A⋮3\)

Tương tự ta có:

\(A\equiv\left(1+-2-3+4\right)\left(mod5\right)\Rightarrow A⋮5\)

Mà 3 và 5 nguyên tố cùng nhau \(\Rightarrow A⋮15\)