From five examples to every number
You test eleven number plates against their reverses. Every difference is a multiple of 99. But there are 900 three-digit numbers. Do all of them work?
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100a + 10b + c. The reverse: 100c + 10b + a. Step 1 of 4
- The number:
100a + 10b + c. The reverse:100c + 10b + a. - Subtract the second line from the first.
- Collect like terms:
99a + 0b − 99c. Thebterms cancel. - The largest number dividing 99 and 99: 99.
The rule
Yes. Eleven plates cannot prove it for all 900 numbers. Letters can.
Write each number as its digits times place value. Do the sum you were asked about. Collect the terms: add the terms that have the same letter. Take out the largest common factor. It is the largest number dividing every term.
Rule: Write both numbers, do the sum, take out the common factor.
No plate was used. The letters stand for all 900 numbers, in four lines.
| Claim | After collecting | Largest common factor |
|---|---|---|
| Two digits, minus reverse | 9a − 9b | 9 |
| Two digits, plus reverse | 11a + 11b | 11 |
| Three digits, minus reverse | 99a + 0b − 99c | 99 |
| Three digits, plus reverse | 101a + 20b + 101c | 1 |
The last factor is 1, so no factor above 1 is promised. 321 + 123 is 444, but 742 + 247 is 989. Only 1 divides both.