In a school, 391 boys and 323 girls have been divided into the largest possible number of equal classes, such that each class has the same number of boys and the same number of girls. How many such equal classes are formed in the school?

Introduction / Context:
This question combines the idea of grouping with the concept of highest common factor. The school wants to divide 391 boys and 323 girls into the largest possible number of equal classes where every class has the same number of boys and the same number of girls. The phrase largest possible number of equal classes hints strongly at finding a divisor common to both groups.

Given Data / Assumptions:
• Number of boys = 391.• Number of girls = 323.• All classes must be equal in composition.• Each class must have the same number of boys as every other class.• Each class must have the same number of girls as every other class.

Concept / Approach:
If we form k equal classes, then the number of boys per class and the number of girls per class must both be integers. Therefore, k must be a common divisor of both 391 and 323. To maximize the number of classes, we must find the greatest common divisor (GCD) of 391 and 323. Once we know the GCD, that value gives the largest possible number of such equal classes.

Step-by-Step Solution:
Step 1: Let the number of classes be k.Step 2: Then boys per class = 391 / k and girls per class = 323 / k, both must be whole numbers.Step 3: Hence, k must divide both 391 and 323.Step 4: Compute the GCD of 391 and 323.Step 5: Find 391 - 323 = 68.Step 6: Now compute GCD(323, 68). Divide 323 by 68: 68 * 4 = 272, remainder 323 - 272 = 51.Step 7: Now find GCD(68, 51). We have 68 - 51 = 17.Step 8: Next, 51 divided by 17 is exact, so GCD(68, 51) = 17.Step 9: Therefore, GCD(391, 323) = 17, which is the maximum possible number of equal classes.

Verification / Alternative check:
If there are 17 classes, then boys per class = 391 / 17 = 23 boys and girls per class = 323 / 17 = 19 girls. Both are whole numbers, and any larger number of classes, such as 19 or 23 or 44, will not divide both numbers exactly. Hence 17 is indeed the largest possible number of classes.

Why Other Options Are Wrong:
23: This divides 391 but does not divide 323 evenly.19: This divides 323 but not 391 without a remainder.44: This divides neither 391 nor 323 exactly, so equal classes cannot be formed.

Common Pitfalls:
Students sometimes mistakenly divide the total number of students instead of considering boys and girls separately, or they look for the least common multiple instead of the greatest common divisor. Remember that to divide each group evenly into the same number of classes, that number must be a common divisor of both group sizes, and to maximize classes we need the greatest such divisor.

Final Answer:
The number of equal classes formed is 17.