Assume that the average of a math test is $90$ points out of $100$ points. If the sum of the scores of all students who took the test is a four digit integer $a95b$,

(1) Find the values of $a$ and $b$

(2) How many students took the test?

I don't even know where to begin. I wonder if there is a way I can utilize the "average" and have the total sum scores divided by the number of students equal $90$, but the only thing I really know (other than that) is it's a four digit number $a95b$. And we don't know the number of students that took the test; so I'm stuck.


The average is 90, so suppose there were $x$ students. Then $90x = a95b$ is the total score. We know $b=0$, since the total score should be divisible by 10. Now, we need to find $a$. An interesting property of $9$ is that for a number to be divisible by $9$, we need to have the sum of its digits be divisible by $9$. Therefore, $a+9+5+0$ must be divisible by $9$. We must then have $a=4$, since $a$ is a single digit. Therefore, the sum of scores is $4950$ and we see that $\frac{4950}{90}=55$ students took the test.

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