2017 AMC 12B Problem 21

Problem 21 of 25HarderAlgebra Number Theory

Last year Isabella took

7

math tests and received

7

different scores, each an integer between

91

and

100,

inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was

95.

What was her score on the sixth test?

Show solution

Solution

Let

S

be the sum of all seven scores. Then

S

is a multiple of

7

with

658 \le S \le 679,

S \in \{658, 665, 672, 679\}.

Since the average after six tests is an integer,

S - 95

is a multiple of

6,

which forces

S = 665.

Then the first six scores sum to

570,

a multiple of

5;

the average after five tests is an integer, so the first five scores also sum to a multiple of

5,

making the sixth score a multiple of

5.

Since all scores differ and the seventh is

95,

the sixth must be

100.

Thus, the correct answer is E.