A Curious Property of Two-Digit Numbers Made from the Same Digits
Statement:
Let x and y be single-digit integers (x, y $\in \{0, 1, \ldots, 9\}$).
Consider two two-digit numbers:
- xy: a number with x in the tens place and y in the units place
- yx: the same digits in reverse order — y in the tens, x in the units
Then the following identity always holds:
xy − yx = 9(x − y)
Proof:
We express the two-digit numbers as:
- xy = 10x + y
- yx = 10y + x
Now subtract:
xy − yx = (10x + y) − (10y + x)
= 10x + y − 10y − x
= 9x − 9y
= 9(x − y)
Example:
Let x = 6, y = 2:
- xy = 62
- yx = 26
- 62 − 26 = 36
- 9(x − y) = 9(6 − 2) = 36
It works!
Remark:
This identity shows that the difference between two-digit numbers made by swapping the same two digits is always divisible by 9, and is directly proportional to the difference of those digits.
Bonus Puzzle:
I thought of two single-digit numbers.
I formed two two-digit numbers by writing them in both possible orders.
The difference between these numbers is 27.
What are the digits?
