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 
tests be 
. Since the mean of her first 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 
integers must be an integer, so the sum of the first test scores must be an multiple of , which implies that the th test score is 
.
Solution 2
First, we find the largest sum of scores which is 
which equals 
. Then we find the smallest sum of scores which is 
which is 
. So the possible sums for the 7 test scores so that they provide an integer average are 
and which are 
and 
respectively. Now in order to get the sum of the first 6 tests, we negate from each sum producing 
and 
. Notice only 
is divisible by so, therefore, the sum of the first tests is . We need to find her score on the 
test so what number minus will give us a number divisible by . Since is the 
test score and all test scores are distinct that only leaves .
Solution 2 (Cheap Solution)
By inspection, the sequences 
and 
work, so the answer is .
Note: A method of finding this "cheap" solution is to create a "mod chart", basically list out the residues of 91-100 modulo 1-7 and then finding the two sequences should be made substantially easier.
See Also
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