Permalink Submitted by Anonymous on October 23, 2012

If first number is 100a + 10b + c
Second number is a+ 10b + 100c
Difference is 99( a-c)

By use of divisibility rules or by exhaustion the 99 times table (1..8) has the property that the outside digits add to 9 and the middle digit is 9

So we have a number of the form 100d+ 90 + f
Add to the digits reversed d +90 + 100f
to get 101(d+f) + 180
And since d + f = 9
The result = 909+ 180 = 1089

Note that this breaks down if a= c e.g. 414 reversed is 414 the difference is zero and digit reversal sum is also zero

The usually stated condition is that the initial three digits are different.
a sufficient but not necessary condition.

## Proof of 1089

If first number is 100a + 10b + c

Second number is a+ 10b + 100c

Difference is 99( a-c)

By use of divisibility rules or by exhaustion the 99 times table (1..8) has the property that the outside digits add to 9 and the middle digit is 9

So we have a number of the form 100d+ 90 + f

Add to the digits reversed d +90 + 100f

to get 101(d+f) + 180

And since d + f = 9

The result = 909+ 180 = 1089

Note that this breaks down if a= c e.g. 414 reversed is 414 the difference is zero and digit reversal sum is also zero

The usually stated condition is that the initial three digits are different.

a sufficient but not necessary condition.

Eg. 441 -144 = 297 =99*3 etc.