# Find odd number with three digits. All digits are different and add up to 12. Difference between 1st two digits = difference between last two digits.

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Let the three digit number be abc

Given that the sum of the digit is 12

==> a + b + c = 12..........(1)

Also, given the difference between the first two digit = the difference between the last two digits:

==> l a-b l = l b-cl

==> (a-b) = (b-c)

==> (a+c) = 2b

Now we will substitue in (1)

==> a+ b + c = 12

==> 2b + b = 12

==> 3b = 12

Now divide by 3:

==> b = 4

==> a+c = 2b= 8

==> a = 8-c ..........(2)

We know that"

(a-b) = (b-c)

==> a > b+c

==> a > 4 + c

==> 8-c > 4+c

==>4 > 2c

==> 2 > c

Since c is an odd integer, then c must be 1

==> b= 4 c = 1 ==> a = 7

**Then the three digit number is 741**

Let x,y and z be the 3 odd number digits.

So x+y +z = 12 ....(1)given

|x-y| = |y-z|.

If x > y> z, then

x-y = y-z .

Therefore 2y = x+z

Put y = (x+z)/2 in x+y+z = 12.

x+(x+z)/2+z = 12.

3(x+z)/2 = 12.

x+z = 24/3 = 8.

x = 5, z= 3 x= 7 and z = 1. and y = 4.

Therefore 543 or 345 , 741 or 147 are possible solutions.

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