Câu hỏi của Đinh Huy Sáng

Question

a=(1+1/2+1/3+1/4+...+1/54).2.3.4.5...54 chứng minh rằng a chia hết cho 55

Kiệt Nguyễn · Accepted Answer

$A=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{54}\right).2.3.4.5...54$

$\Rightarrow A=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{54}\right).2.3.4.5...11.12...54$

$\Rightarrow\hept{\begin{cases}A⋮5\A⋮11\end{cases}}$mà $\left(5,11\right)=1$ nên $A⋮55\left(đpcm\right)$

Câu trả lời:

$A=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{54}\right).2.3.4.5...54$

$\Rightarrow A=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{54}\right).2.3.4.5...11.12...54$

$\Rightarrow\hept{\begin{cases}A⋮5\A⋮11\end{cases}}$mà $\left(5,11\right)=1$ nên $A⋮55\left(đpcm\right)$