Problem 29 · 2019 Math Kangaroo
Stretch
Number Theory
place-valuecasework
The numbers a, b and c are three-digit numbers, and in each number the first digit is equal to the last one. Furthermore \(b = 2a + 1\) and \(c = 2b + 1\). How many possible values are there for the number a?
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Answer: C — 2
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Hint 1 of 2
“First digit equals last” means each of a, b, c is a 3-digit number of the form x?x.
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Hint 2 of 2
Use b = 2a + 1 and c = 2b + 1 and test which starting a keep all three in that form.
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Approach: chase the doubling chain through 3-digit ‘x?x’ numbers
- a, b, c each read x?x (first digit = last). With b = 2a + 1 and c = 2b + 1, only a few a work.
- a = 181 gives b = 363, c = 727; a = 191 gives b = 383, c = 767 — both valid.
- No other a works, so there are 2 possible values.
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