Three digits, reversed and subtracted
A number plate reads 742. Its reverse is 247: the digits written backwards. 742 − 247 = 495. Plate 631 gives 631 − 136, also 495. Why is it the same?
Loading lab…
- 742 is three digit boxes: 7, 4, 2.
- Reverse them: 2, 4, 7. The middle box did not move.
- 742 − 247 = 495.
- 495 = 99 × 5, and 5 is the gap between 7 and 2.
The rule
In both plates, the first and last digits are 5 apart. So both give 99 × 5 = 495.
Call the digits a, b and c. Here a is the hundreds digit. b is the tens digit. c is the ones digit. The number is 100a + 10b + c. Its reverse is 100c + 10b + a.
Take a bigger than c, so the number is the bigger one. Subtract the reverse from it. Both have 10b, so the b terms cancel. What is left is 99a − 99c, which is 99(a − c).
The outer digits are a and c. Their gap is a − c.
Rule: Bigger minus smaller: the middle digit drops out. The difference is 99 times the outer gap.
| Number | Reverse | Difference | a − c | 99 × (a − c) |
|---|---|---|---|---|
| 321 | 123 | 198 | 2 | 198 |
| 391 | 193 | 198 | 2 | 198 |
| 742 | 247 | 495 | 5 | 495 |
| 504 | 405 | 99 | 1 | 99 |
Rows 1 and 2 share outer digits but not the middle digit. The answer is the same.
99 is 9 × 11. So every difference here is a multiple of 9 and of 11.