2017 AMC 10B Problems/Problem 25
Problem
Last year Isabella took math tests and received
different scores, each an integer between
and
, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was
. What was her score on the sixth test?
Solution 1
Let the sum of the scores of Isabella's first 6 tests be . Since the mean of her first 7 scores is an integer, then
, or
. Also,
, so by CRT,
. We also know that
, so by inspection,
. However, we also have that the mean of the first 5 integers must be an integer, so the sum of the first 5 test scores must be an multiple of 5, which implies that the 6th test score is
.
Cheap Solution
By inspection, the sequences and
work, so the answer is
.