2017 AMC 10B Problems/Problem 1
Mary thought of a positive two-digit number. She multiplied it by and added . Then she switched the digits of the result, obtaining a number between and , inclusive. What was Mary's number?
Let her -digit number be . Multiplying by makes it a multiple of , meaning that the sum of its digits is divisible by . Adding on increases the sum of the digits by (we can ignore numbers such as ) and reversing the digits keeps the sum of the digits the same; this means that the resulting number must be more than a multiple of . There are two such numbers between and : and Now that we have narrowed down the choices, we can simply test the answers to see which one will provide a two-digit number when the steps are reversed: For we reverse the digits, resulting in Subtracting , we get We can already see that dividing this by will not be a two-digit number, so does not meet our requirements. Therefore, the answer must be the reversed steps applied to We have the following: Therefore, our answer is .
Working backwards, we reverse the digits of each number from ~ and subtract from each, so we have The only numbers from this list that are divisible by are and . We divide both by , yielding and . Since is not a two-digit number, the answer is .
You can just plug in the numbers to see which one works. When you get to , you multiply by and add to get . When you reverse the digits of , you get , which is within the given range. Thus, the answer is .
Solution 4 (Fastest Way)
Let x be the original number. The last digit of must be so the last digit of must be . The only answer choice that satisfies this is .
Subtract from the numbers through . This yields , , , , and . Of these, the only ones divisible by are and . Therefore, the only possible values are and . Switching the digits of each, we get and . Subtracting from each, we get the numbers and . Dividing each by , we get and . The only two-digit number is , so the answer is .
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