Problem 13 · 2021 Math Kangaroo
Hard
Number Theory
place-valuedivisibility
How many three-digit natural numbers have the property that when their digits are written in reverse order, the result is a three-digit number which is 99 more than the original number?
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Answer: D — 80
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Hint 1 of 2
Write the number and its reversal using place value and subtract.
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Hint 2 of 2
The difference depends only on the first and last digits; set it equal to 99.
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Approach: use place value to relate a number and its reversal
- For a 3-digit number 100a+10b+c, the reversal is 100c+10b+a, and their difference is 99(c − a).
- Setting 99(c − a) = 99 gives c = a + 1.
- With a from 1 to 8 (so c stays a digit) and b any of 0–9, that is 8 × 10 = 80 numbers.
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