Question

The number of positive even divisors of 6300 is:

• A. \( 30 \)
• B. \( 24 \)
• C. \( 18 \)
• D. \( 36 \)

Hint:

To find even divisors, compute total divisors and subtract those that exclude the factor of 2.

Answer 1

The Correct Option is D

Step 1: Prime factorize 6300
\[ 6300 = 2^2 \cdot 3^2 \cdot 5^2 \cdot 7 \]
Step 2: Total divisors
Total number of positive divisors: \[ (2 + 1)(2 + 1)(2 + 1)(1 + 1) = 3 \cdot 3 \cdot 3 \cdot 2 = 54 \]
Step 3: Count only even divisors
Even divisors must include at least one factor of 2. So remove all divisors that do not include 2: \[ \text{Odd divisors (excluding 2)} = (0 + 1)(2 + 1)(2 + 1)(1 + 1) = 1 \cdot 3 \cdot 3 \cdot 2 = 18 \] Thus: \[ \text{Even divisors} = 54 - 18 = 36 \]