- #1

- 690

- 180

- Homework Statement
- Without performing the divisions, determine whether the integer ## 1010908899 ## is divisible by ## 7, 11 ##, and ## 13 ##.

- Relevant Equations
- None.

Consider the integer ## 1010908899 ##.

Observe that ## 7\cdot 11\cdot 13=1001 ##.

Then ## 10^{3}\equiv -1\pmod {1001} ##.

Thus

\begin{align*}

&1010908899\equiv (1\cdot 10^{9}+10\cdot 10^{6}+908\cdot 10^{3}+899)\pmod {1001}\\

&\equiv (-1+10-908+899)\pmod {1001}\\

&\equiv 0\pmod {1001}.\\

\end{align*}

Therefore, the integer ## 1010908899 ## is divisible by ## 7, 11 ##, and ## 13 ##.

Observe that ## 7\cdot 11\cdot 13=1001 ##.

Then ## 10^{3}\equiv -1\pmod {1001} ##.

Thus

\begin{align*}

&1010908899\equiv (1\cdot 10^{9}+10\cdot 10^{6}+908\cdot 10^{3}+899)\pmod {1001}\\

&\equiv (-1+10-908+899)\pmod {1001}\\

&\equiv 0\pmod {1001}.\\

\end{align*}

Therefore, the integer ## 1010908899 ## is divisible by ## 7, 11 ##, and ## 13 ##.