# The number of 6 digit numbers that can be formed using the digits 0,1,2,5,7 and 9 which are divisible by 11 and no digit is repeated, is:

### Divisibility Rules for 11

If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11 completely.

In order to check whether a number like 2143 is divisible by 11, below is the following procedure.

• Group the alternative digits i.e. digits which are in odd places together and digits in even places together. Here 24 and 13 are two groups.
• Take the sum of the digits of each group i.e. 2+4=6 and 1+3= 4
• Now find the difference of the sums; 6-4=2
• If the difference is divisible by 11, then the original number is also divisible by 11. Here 2 is the difference which is not divisible by 11.
• Therefore, 2143 is not divisible by 11.

### Solution

Given digits: 0,1,2,4,5,7,9

Sum of the given digit is 0+1+2+5+7+9 = 24

Let the digits be abcdef

The numbers abcdef is divisible by 11 if

For a number to be divisible by 11 , difference of sum of alternate digits of the number should be of the form

|(a+c+e )-(b+d+f)is a multiple of 11

a+c+e=b+d+f=12

Case 1: {a,c,e}={7,5,0}

{b,d,f}={9,2,1}

So, 2 x 2! x 3! = 24

Case 2: {a,c,e}={9 , 2, 1}

{b,d,f}={7, 5, 0}

So, 3! x 3! = 36

Total = 24 + 36 = 60

Articles to explore