- #1
Math100
- 768
- 219
- 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 ##.