Distribution using H.C.F: An inspector of schools wants to distribute 84 balls and 180 bats equally among as many boys as possible, with each boy receiving the same number of balls and the same number of bats. What is the greatest possible number of boys?

Introduction / Context:
This is a classic highest common factor (H.C.F) application. To distribute two different items equally among the maximum number of recipients, the number of recipients must divide each item count exactly. Therefore, we seek the largest number that divides both 84 (balls) and 180 (bats).

Given Data / Assumptions:

• Balls = 84
• Bats = 180
• Each boy must get the same number of balls and the same number of bats.
• We need the greatest possible number of boys.

Concept / Approach:
The greatest possible number of boys equals HCF(84, 180). Factorization or Euclidean Algorithm may be used. If HCF = k, then both 84 and 180 are multiples of k, allowing equal distribution without remainder.

Step-by-Step Solution:

Verification / Alternative check:
Using the Euclidean Algorithm: HCF(180, 84) ⇒ 180 mod 84 = 12; 84 mod 12 = 0 ⇒ HCF = 12. With 12 boys, each gets 84/12 = 7 balls and 180/12 = 15 bats (both integers).

Why Other Options Are Wrong:
16, 15, 14, and 18 do not divide both 84 and 180 exactly, so equal distribution would fail for at least one item.

Common Pitfalls:
Using LCM instead of HCF, or checking divisibility against only one of the item counts. Both counts must be divisible.

Final Answer:
12