Section Average Marks and Student Difference

Let $n$ be the number of students in section A. Then section B has $n + 10$ students. Total students $= 2n + 10$. Combined average $A_{avg}$: $A_{avg} = \frac{32n + 60(n + 10)}{2n + 10} = \frac{92n + 600}{2n + 10} = 46 + \frac{140}{2n + 10}$

For $A_{avg}$ to be an integer, $\frac{140}{2n + 10}$ must be an integer, which means $2n + 10$ must be a divisor of 140. Since $n \ge 1$, $2n + 10 \ge 12$, and $2n + 10$ is even.

Divisors of 140 that are even and $\ge 12$: $14, 20, 28, 70, 140$.

Corresponding values of $2n + 10$:

• $2n + 10 = 14 \implies n = 2$
• $2n + 10 = 20 \implies n = 5$
• $2n + 10 = 28 \implies n = 9$
• $2n + 10 = 70 \implies n = 30$
• $2n + 10 = 140 \implies n = 65$

Maximum possible $n = 65$. Minimum possible $n = 2$.

Difference $= 65 - 2 = 63$.