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2002 AMC 12B Problem 10

Problem 10 of 25EasierNumber TheoryCounting & Probability

How many different integers can be expressed as the sum of three distinct members of the set {1,4,7,10,13,16,19}?\{1,4,7,10,13,16,19\}?

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Solution

Every element is one more than a multiple of 3,3, so any sum of three of them is a multiple of 3.3. The smallest sum is 1+4+7=121+4+7=12 and the largest is 13+16+19=48,13+16+19=48, and every multiple of 33 between them is attainable. There are 1313 multiples of 33 from 1212 to 48.48. Thus, the correct answer is A.

More practice

Concepts: modular arithmetic · counting integers in a range

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