A school with 862 students will participate in a scavenger hunt, whose rules are:

I) The number of registrants must be a number between $\frac{2}{3}$ and $\frac{7}{9}$ of the total number of students in the school.

II) Since the students will be divided into groups of 11, the number of registrants must be a multiple of 11.

How many are the possible numbers of registrants for this school?

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Solution

Since $\frac{2}{3} \cdot 862=574, \overline{6}$ and $\frac{7}{9} \cdot 862=670, \overline{4}$, the number of students must be a multiple of 11 in the interval $[575,670]$. The smallest multiple in the interval is $53 \cdot 11=583$ and the largest is $60 \cdot 11=660$. Therefore, there are $60-52=8$ possible different numbers of students from this school participating in the scavenger hunt.