2000 AMC 10 Problem 1

Problem 1 of 25EasierAlgebra Number Theory

In the year

2001,

the United States will host the International Mathematical Olympiad. Let

I,

M,

and

O

be distinct positive integers such that the product

I \cdot M \cdot O = 2001.

What is the largest possible value of the sum

I + M + O?

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Solution

Factoring gives

2001 = 3 \cdot 23 \cdot 29.

If one factor is

1,

the possible pairs for the other two factors are

(3,667),

(23,87),

and

(29,69).

Their corresponding sums with

1

are

671,

111,

and

99.

(The pair

(1,2001)

would repeat the factor

1.

) If no factor is

1,

all three prime factors must be split among the three integers, giving only

3,23,29

and a much smaller sum. Therefore, the largest possible sum is

1 + 3 + 667 = 671.

Thus, the correct answer is E.