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?

$71$

$76$

$80$

$82$

$91$

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.

2000 AMC 10 Problem 14

Solution:

Problem 14 in Other Years