Question

# Write a seven digit number without repeating any digit which is divisible by $2,3,4,6,8,9,11$

Hint:The seven digit number starts with $10,00,000$ and ends with $99,99,999$ , hence our answer is hidden in between this. First we find the least common multiple for $2,3,4,6,8,9,11$. It is essential to find a common multiple; but we cannot check if a seven digit number is divisible by $2,3,4,6,8,9,11$ instead of using a common multiple for checking. Both methods will be the same.

It is given that the set of numbers are $2,3,4,6,8,9,11$
We have to find the seven digit number is divisible by each of $2,3,4,6,8,9,11$
First we find the least common multiple for the set of given numbers.
Here ${2^3} \times 9 \times 11 = 792$ is the least common multiple for the set of given numbers.

First we take the least seven digit number $10,00,000$ is divisible by $792$
Then, we get quotient $= 1262$ and remainder $= 496$

Here, we are trying to find the least seven digit number which is divisible by $792$
$\therefore 792 \times 1262 = 999504$ but it is greatest six digit number which divisible by $792$
Also $10,00,000$ is not perfectly divide by $792$ , where $792 \times 1262 = 999504$ is greatest six digit number if we add by $792$ , it is the first seven digit number divisible by $792$ .

Add $999504 + 792 = 1000296$ it is the least seven digit number which is divisible by $792$,
Suppose we take a greatest seven digit number $99,99,999$ divisible by $792$
Then we get, quotient=$12626$, remainder $= 207$
Also $99,99,999$ is not perfectly dividing by $792$, so either we can subtract $99,99,999$ by $207$ or multiply $12626$ by$792$.

Therefore, $792 \times 12626 = 9999792$ it is the greatest seven digit number which is divisible by $792$
Hence the set of seven digit number which is divisible by $792 = \{ 1000296,1001088,....9999792\}$

From the set the number we obtained $792 \times 1307 = 1035144$ .

Note: It is not the only solution for this sum; $792 \times 5184 = 4105728$, it is also one of the solutions.It has thousands of multiples in the set which is divisible by $792$ means the number is divisible by $2,3,4,6,8,9,11$.