2000 AMC 12 Problem 9

Question

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?

71

76

80

82

91

Answer: C

Concepts:divisibility modular arithmetic mean

Difficulty rating: 1580

Small hint:

The sum of the first

k

scores entered must be divisible by

k

Big hint:

Use divisibility by

3

then by

4

; the residues of the scores modulo

3

are

2, 1, 2, 1, 1

Solution:

The total is

71 + 76 + 80 + 82 + 91 = 400,

which is divisible by

5.

The sum of the first three scores must be divisible by

3.

Modulo

3,

the scores are

2, 1, 2, 1, 1.

The only triple summing to a multiple of

3

is

76 + 82 + 91 = 249,

so these are the first three (with

91

third, since the first two,

76

and

82,

must have equal parity).

Since

249 \equiv 1 \pmod 4,

the fourth score must be

\equiv 3 \pmod 4,

which is

71.

That leaves

80

as the last score entered.

Thus, the correct answer is C.