2014 AMC 12B Problems/Problem 10
Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, miles was displayed on the odometer, where is a 3-digit number with and . At the end of the trip, the odometer showed miles. What is .
We know that the number of miles she drove is divisible by , so and must either be the equal or differ by . We can quickly conclude that the former is impossible, so and must be apart. Because we know that and and , we find that the only possible values for and are and , respectively. Because , . Therefore, we have
Let the number of hours Danica drove be . Then we know that = . Simplifying, we have , or . Thus, k is divisible by . Because , must be , and therefore . Because and , , and , and our answer is , or .
Solution 3 (Testing Values)
We know that since adding a multiple of will increase the number, we know that . After testing values for , , and , adding their squares, we see that using non-zero integers as the digits can yield a maximum value of as 27, specifically the combination of . However, we see that no multiple of can be added to to achieve . No combination of non-zero integers can create a higher answer, so we must try testing with a zero. Realizing that only can be zero, we first try to see if (which creates 26 from the expression , answer choice A) can be added to a multiple of to create , which we find it cannot. However, we quickly realize that the next consecutive pair does satisfy this condition, and . We come to this quickly after realizing that if the multiple of was an even multiple, it would have to end in and thus the digit would remain unchanged, so it must be an odd multiple, which will carry over the digit, so .
-Solution by Joeya
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