A number minus its reverse

The idea

A two-digit number ab equals 10a + b and its reverse ba equals 10b + a, so their difference, (10b + a) – (10a + b) = 9(b – a), is always divisible by 9.

Mukta's trick

Choose a two-digit number with two different digits. Reverse the digits. Find the difference. Divide it by 9. There is no remainder!

For 47: 74 − 47 = 27 and 27 ÷ 9 = 3.

Will it work for every such number?

• Yes, there is never a remainder
• Only for 47
• Only for some numbers

Trade cubes for rods

A rod is worth 10, a cube is worth 1. The number 47 is 4 rods and 7 cubes. Trade a cube for a rod again and again until the number turns into its reverse. Watch what each trade does. Change the digits with ▲ ▼.

Nine for every step

A two-digit number ab equals 10a + b and its reverse ba equals 10b + a, so their difference, (10b + a) – (10a + b) = 9(b – a), is always divisible by 9.

Each trade gives away 1 cube (−1) for 1 rod (+10), so the number grows by 9. Going from ab to ba takes b − a trades. That is 9 × (b − a).

In letters: (10b + a) − (10a + b) = 10b + a − 10a − b = 9b − 9a = 9(b − a)

For 47: 74 − 47 = 27 = 9 × 3 and 7 − 4 = 3. Divide by 9 and you get the gap between the digits.

If the tens digit is the bigger one, the reverse is the smaller number, and the difference is 9(a − b). It is still divisible by 9.

Notes

A two-digit number ab equals 10a + b and its reverse ba equals 10b + a, so their difference, (10b + a) – (10a + b) = 9(b – a), is always divisible by 9.

Check yourself

The digits of a number are 3 and 8. Take the bigger of the number and its reverse, subtract the smaller, and divide by 9. What do you get?

Answer: 5

83 − 38 = 45 and 45 ÷ 9 = 5. The gap between the digits 8 and 3 is 5. So 83 − 38 = 9 × 5.

Which of these can not be the difference between a two-digit number and its reverse?

Simplify (10b + a) − (10a + b).

Mukta picks 55, with both digits the same. Does the trick still leave no remainder?

• 27. It can: 74 − 47 = 27 = 9 × 3.
• 50 — correct. Right! The difference is 9 × (the gap between the digits), a multiple of 9. 50 is not: 9 × 5 = 45 and 9 × 6 = 54.
• 45. It can: 83 − 38 = 45 = 9 × 5.
• 72. It can: 91 − 19 = 72 = 9 × 8.
• 11b − 11a. Take care with the minus: it applies to both 10a and b. So 10b − b = 9b, not 11b.
• 10b − 10a. The a and the b of the second bracket must be taken away too: 10b − b = 9b and a − 10a = −9a.
• 9b − 9a — correct. Yes! 10b − b = 9b and a − 10a = −9a. So the answer is 9b − 9a = 9(b − a).
• 9a − 9b. That is ab minus ba, the other way round. Here ba is the first number, so we get 9b − 9a.
• Yes, the difference is 0 and 0 is divisible by 9 — correct. Yes! Here b − a = 0, so 9(b − a) = 9 × 0 = 0, which is divisible by 9.
• No, 55 cannot be reversed. 55 reversed is 55. The two numbers are equal and the difference is 0.
• Yes, the answer is 9. The difference is 55 − 55 = 0, not 9.