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2006 AMC 12A Problem 4

Problem 4 of 25EasierAlgebraNumber Theory

A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?

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Solution

The two minutes digits sum to at most 5+9=14,5 + 9 = 14, at 5959 minutes past the hour. For the hour, a single digit 99 gives digit sum 9,9, which beats any two-digit hour (10,11,12(10, 11, 12 give at most 1+2=3).1 + 2 = 3). The largest total is 14+9=23,14 + 9 = 23, occurring at 9 ⁣: ⁣59.9\!:\!59. Thus, the correct answer is E.

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Concepts: digits · clock

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