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

[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 $13$.

 

[malawi_glenn]

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

 

[Math100]

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.

 

[malawi_glenn]

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.

 

[fresh_42]

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.

 

[WWGD]

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]

I was thinking about adding spacing $1\,010\,908\,899$ for "ocular ease" :)