3. In a school, there are fewer than 400 students in the sixth grade, divided into several parallel classes. Six of the parallel classes have the same number of students, and these six classes together have more than 150 students. In the remaining parallel classes in the sixth grade, there are 15% more students in total than in these six classes combined. How many students are there in total in the sixth grade in this school?
Solution. Let's denote with $n$ the total number of students in six parallel classes. Thus, 6|n. From the fact that in the other classes there are 15% more students in total than in these six parallel classes, we have that $15 \% \cdot n=\frac{15}{100} \cdot n=\frac{3}{20} \cdot n$ must be a natural number, from which it follows that $20 \mid n$. From $6 \mid n$ and $20 \mid n$ it follows that $60 \mid n$, that is, $n=60k$. From the condition of the problem, we have that $n>150$ and $n+n+15 \% \cdot n=2.15 n<400$. The smallest natural number divisible by 60 and greater than 150 is $n=180$ and for it we have that $2.15 n=2.15 \cdot 180=387$. The next natural number divisible by 60 is $n=240$, but for $n \geq 240$ we have that $2.15 n \geq 2.15 \cdot 240=516$, so the condition $2.15 n<400$ is not met. Therefore, the total number of students in the sixth grade in this school is 387 students.