# Determine whether the integer ## 1010908899 ## is divisible by....

• Math100

#### Math100

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 ##.

Do you have a question somewhere? What do you need help with?

Do you have a question somewhere? What do you need help with?
The question is written in the homework statement. I just wanted someone to verify/confirm that my work is correct/accurate.

The question is written in the homework statement.
Yeah I could see that, but what is your question to us?
I just wanted someone to verify/confirm that my work is correct/accurate.
Ok.

Looks good to me.
You can include:
##10^9 \equiv_{1001} (10^3)^3 \equiv_{1001}(-1)^3 \equiv_{1001} -1##
##10^6 \equiv_{1001} (10^3)^2 \equiv_{1001}(-1)^2 \equiv_{1001} 1##
for the sake of completness.

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 ##.
This is correct, although I'm not sure whether you are supposed to solve it like that or apply the rules for divisibility by ##7,11,13.## IIRC then there are rules. But your solution is nicer.

Math100
Maybe if you add ##1,001## to your original, it may become more clear
## 1,010, 908,899 +1,001=1,010,909,900-1,001,000,000=9,909,900##

malawi_glenn