An inspector of school wishes to distribute 84 balls and 180 bat equally among a number of boys. Find the greatest number receiving the gift in this way-

Options

A. 14
B. 15
C. 16
D. 12

Answer

To solve the problem of distributing 84 balls and 180 bats equally among a number of boys, we need to find the greatest number of boys that can receive the gifts. This can be done by finding the highest common factor (HCF) of the two numbers.

### Step-by-Step Solution:

**Step 1: Find the prime factorization of 84.**
- To find the prime factors of 84, we divide it by the smallest prime number (2) and continue the process:
- 84 ÷ 2 = 42
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7
- 7 is a prime number.
- Therefore, the prime factorization of 84 is:
\[ 84 = 2^2 \times 3^1 \times 7^1 \]

**Step 2: Find the prime factorization of 180.**
- Similarly, we find the prime factors of 180:
- 180 ÷ 2 = 90
- 90 ÷ 2 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 is a prime number.
- Thus, the prime factorization of 180 is:
\[ 180 = 2^2 \times 3^2 \times 5^1 \]

**Step 3: Identify the common prime factors.**
- From the factorizations:
- For 84: \(2^2, 3^1, 7^1\)
- For 180: \(2^2, 3^2, 5^1\)
- The common prime factors are \(2\) and \(3\).

**Step 4: Determine the lowest power of each common prime factor.**
- For \(2\): The lowest power is \(2^2\).
- For \(3\): The lowest power is \(3^1\).
- There is no common factor for \(7\) and \(5\).

**Step 5: Calculate the HCF.**
- The HCF is found by multiplying the lowest powers of the common prime factors:
\[ HCF = 2^2 \times 3^1 = 4 \times 3 = 12 \]

**Step 6: Conclusion.**
- The greatest number of boys that can receive the gifts equally is **12**.