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2000 AMC 12 Problem 1

Problem 1 of 25EasierNumber TheoryProblem-Solving Techniques

In the year 2001,2001, the United States will host the International Mathematical Olympiad. Let I,I, M,M, and OO be distinct positive integers such that the product I⋅M⋅O=2001.I \cdot M \cdot O = 2001. What is the largest possible value of the sum I+M+O?I + M + O?

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Solution

Factoring gives 2001=3⋅23⋅29.2001 = 3 \cdot 23 \cdot 29. To maximize the sum of three distinct positive integers with this product, take I=1,I = 1, M=3,M = 3, and O=23⋅29=667.O = 23 \cdot 29 = 667. The largest sum is 1+3+667=671.1 + 3 + 667 = 671. Thus, the correct answer is E.
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Tagged: prime factorization · optimization

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