# 2017 AMC 12B Problems/Problem 21

## Problem 21

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? $\textbf{(A)}\ 92\qquad\textbf{(B)}\ 94\qquad\textbf{(C)}\ 96\qquad\textbf{(D)}\ 98\qquad\textbf{(E)}\ 100$

## Solution 1

Let us simplify the problem. Since all of Isabella's test scores can be expressed as the sum of $90$ and an integer between $1$ and $10$, we rewrite the problem into receiving scores between $1$ and $10$. Later, we can add $90$ to her score to obtain the real answer.

From this point of view, the problem states that Isabella's score on the seventh test was $5$. We note that Isabella received $7$ integer scores out of $1$ to $10$. Since $5$ is already given as the seventh test score, the possible scores for Isabella on the other six tests are $S={1,2,3,4,6,7,8,9,10}$.

The average score for the seven tests must be an integer. In other words, six distinct integers must be picked from set $S$ above, and their sum with $5$ must be a multiple of $7$. The interval containing the possible sums of the six numbers in S are from $1 +2+3+4+6+7=23$ to $4+6+7+8+9+10=44$. We must now find multiples of $7$ from the interval $23+5 = 28$ to $44+5=49$. There are four possibilities: $28$, $35$, $42$, $49$. However, we also note that the sum of the six numbers (besides $5$) must be a multiple of $6$ as well. Thus, $35$ is the only valid choice.(The six numbers sum to $30$.)

Thus the sum of the six numbers equals to $30$. We apply the logic above in a similar way for the sum of the scores from the first test to the fifth test. The sum must be a multiple of $5$. The possible interval is from $1+2+3+4+6=16$ to $6+7+8+9+10=40$. Since the sum of the five scores must be less than $30$, the only possibilities are $20$ and $25$. However, we notice that $25$ does not work because the seventh score turns out to be $5$ from the calculation. Therefore, the sum of Isabella's scores from test $1$ to $5$ is $20$. Therefore, her score on the sixth test is $10$. Our final answer is $10+90= \boxed{\textbf{(E) }100}$.

## Solution 2

Let $n$ be Isabella's average after $6$ tests. $6n+95 \equiv 0 \pmod{7}$, so $n \equiv 4 \pmod{7}$. The only integer between $90$ and $100$ that satisfies this condition is $95$. Let $m$ be Isabella's average after $5$ tests, and let $a$ be her sixth test score. $5m+a= 6\cdot95 \equiv 0 \pmod{5}$, so $a$ is a multiple of $5$. Since $100$ is the only choice that is a multiple of $5$, the answer is $\boxed{\textbf{(E) }100}$.

## Solution 3

Let $T$ be the total sum of Isabella's first five test scores, and let $S$ be her score on the sixth test. It follows that $T\equiv 0\pmod {5}$, $T+S\equiv 0\pmod {6}$, and $T+S+95\equiv 0\pmod {7}$, since at each step, her average score was an integer. Using the last equivalence, $T+S\equiv -95\equiv 3\pmod{7}$, so we have a system of equivalences for $T+S$. Solving this using the Chinese Remainder Theorem, we get $T+S\equiv (0)(7)(1)+(3)(6)(-1) \equiv -18 \equiv 24 \pmod{42}$.

Now let's put a bound on $T$. Using the given information that each test score was a distinct integer from $91$ to $100$ inclusive and that the seventh score was 95, we get $91+92+93+94+96\leq T\leq 100+99+98+97+96$. Since $T\equiv 0 \pmod{5}$, we get $T=470,475,480,485,490$. Therefore, $T\equiv 8,13,18,23,28\pmod{42}$

The last preparation step will involve calculating all the possible test scores $\pmod {42}$. Here they are: $91\equiv 7\pmod{42},92\equiv 8\pmod{42},93\equiv 9\pmod{42},94\equiv 10\pmod{42},95\equiv 11\pmod{42},96\equiv 12\pmod{42},97\equiv 13\pmod{42},98\equiv 14\pmod{42},99\equiv 15\pmod{42},100\equiv 16\pmod{42}$. This means that $S\equiv 7,8,9,10,12,13,14,15,16\pmod{42}$. Note that $11$ is not in the previous list because it corresponds to a score of $95$, which we cannot have.

We must have $T+S\equiv 24\pmod{42}$, and using the possible values we found for $T$ and $S$, the only two that sum to $24$ are $T\equiv 8\pmod{42}$ and $S\equiv 16\pmod{42}$. This corresponds to an $S$ value of $100$, so the answer is $\boxed{\bold{(E)} 100}$.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 