Why a Number Minus Its Reverse Is a Multiple of 9

Why is a two-digit number minus its reverse always a multiple of 9?

Subtracting the reverse of a two-digit number always gives a multiple of 9. Take 82 − 28 = 54, which is 9 × 6. Split both numbers into tens and units: 82 is 80 + 2, and 28 is 20 + 8.

The 8 is worth 80 in the first number and 8 in the second, so it contributes 80 − 8 = 72. The 2 is worth 2 in the first number and 20 in the second, so it takes away 20 − 2 = 18. The total is 72 − 18 = 54, which is 9 × 8 − 9 × 2, or 9 × (8 − 2).

So the answer is always 9 times the difference of the digits, and you can find it without subtracting: the digits of 93 are 6 apart, so 93 − 39 = 9 × 6 = 54. When the digits are equal, as in 44, the answer is 0, which is 9 × 0.