Question

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

Answer 1

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.

Complete step-by-step answer:
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 \[2,3,4,6,8,9,11\].