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

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.

2000 AMC 12 Problem 9

Answer: C

Difficulty rating: 1580

Solution:

Problem 9 in Other Years