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2017 AMC 10B Problem 1

Problem 1 of 25EasierNumber Theory

Mary thought of a positive two-digit number. She multiplied it by 33 and added 11.11. Then she switched the digits of the result, obtaining a number between 7171 and 75,75, inclusive. What was Mary’s number?

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Solution

Work backward from the possible results. Reversing 71,72,73,74,7571,72,73,74,75 gives 17,27,37,47,57,17,27,37,47,57, respectively. Subtracting 1111 gives 6,16,26,36,46.6,16,26,36,46. Only 66 and 3636 are divisible by 3,3, and only 36÷3=1236\div3=12 is a two-digit number. Thus, the correct answer is B .

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Concepts: digits · divisibility · work backwards

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.