The Mysterious Number 22

Select any 3-digit number with all digits different from one another. Write all possible 2-digit numbers that can be formed from the 3-digits selected earlier. Then divide their sum by the sum of the digits in the original 3-digit number.

You should always get the same answer, 22. There ought to be a big resulting “Wow!”

For example, consider the three-digit number 365.

Take the sum of all the possible two-digit numbers that can be formed from these three digits:

 
36+35+63+53+65+56 = 308.

The sum of the digits of the original number is
 3 + 6 + 5 = 14.

Then 308/14 = 22.

Math behind this:

To analyze this unusual result, we will begin with a general representation of the number: 100x + 10y + z.

We now take the sum of all the two-digit numbers taken from the three digits:

(10x+y)+ (10y+x) + (10x+z) + (10z+x) + (10y+z) + (10z+y)

=10(2x+2y+2z) + (2x+2y+2z)

=11(2x+2y+2z)

=22(x+y+z)

which, when divided by the sum of the digits, (x + y + z), is 22.

These illustrations show the value of algebra in explaining simple arithmetic phenomena.