Why a Number Minus Its Reverse Is a Multiple of 9

Algebraic shortcut: ab - ba = 9(a-b)

82 − 28 = 54

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.

  1. 82 − 28the same two digits, swapped(80 + 2) − (20 + 8)

    Split both: 82 is 80 + 2, and 28 is 20 + 8 — the same digits, swapped.

  2. (80 − 8) + (2 − 20)each digit gains a column and loses one9 × 8 − 9 × 2

    The 8 gains 80 and loses 8: nine 8s. The 2 loses 20 and gains 2: nine 2s off.

  3. 9 × (8 − 2)9 × 6 = 54

    So subtract the digits and multiply by 9: 8 − 2 = 6, and 9 × 6 = 54.

Your turn

Three to try in your head — tap what you get.

41 - 14

61 - 16

76 - 67

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