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2006 AMC 10A 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 minutes run from 0000 to 59,59, so the largest digit sum for the minutes is 5+9=14,5 + 9 = 14, at 5959 minutes. For the hour, the single digit 99 beats 1+2=31 + 2 = 3 from 12.12. The largest total is 9+14=23,9 + 14 = 23, occurring at 9 ⁣: ⁣59.9\!:\!59. Thus, the correct answer is E.

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Concepts: clock · digits · optimization

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