Question

Explain how they work?
Write down any three-digit number(for example, $425$). Make a six-digit number by repeating these digits in the same order$\left(425425\right)$. Your new number is divisible by $7,11$, and $13$.

Answer 1

Note that $7\times 11\times 13=1001$
Take any three digit number say, $abc$.

Then $abc\times 1001=abcabc$

Therefore, the six digit number $abcabc$ is divisible by $7,11$ and $13$

According to the Question

$\Rightarrow$ Let a 3 digit number be $abc$

Now repeating it we get $abcabc$

On Dividing $abcabc$ by $abc$ we get $1001$

$\Rightarrow abc\times 1001=abcabc$

Therefore $1001$ is factor of $abcabc$ and $1001=13\times 7\times 11$

Therefore $abcabc$ is divisible by $7,11$ and $13$