2000 AMC 10 Problem 14

Problem 14 of 25IntermediateAlgebra Number Theory

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were

71,

76,

80,

82,

and

91.

What was the last score Mrs. Walter entered?

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Solution

The residues of

71, 76, 80, 82, 91

modulo

3

are

2, 1, 2, 1, 1.

The sum of the first three scores must be divisible by

3,

and the only such triple is

76 + 82 + 91 = 249,

so the third score entered is

91

and the first two are

76

and

82.

Since

249

is one more than a multiple of

4,

the fourth score must be three more than a multiple of

4,

which only

71

satisfies. That leaves

80

as the fifth and last score. Indeed

76, 158, 249, 320, 400

are divisible by

1, 2, 3, 4, 5.

Thus, the correct answer is C.

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