Question

Total number of even divisors of \(2079000\) which are divisible by \(15\) are:

• A. \(54\)
• B. \(128\)
• C. \(108\)
• D. \(72\)

Hint:

Apply conditions first, then count valid exponent choices.

Answer 1

The Correct Option is C

Concept:

Step 1: Prime factorization: \[ 2079000 = 2^3 \cdot 3^3 \cdot 5^3 \cdot 7 \cdot 11 \]

Step 2: Conditions:
•Even $\Rightarrow$ at least one factor of \(2\)
•Divisible by 15 $\Rightarrow$ must include \(3\) and \(5\)

Step 3: Choices:
•\(2^1,2^2,2^3\) $\Rightarrow$ 3 ways
•\(3^1,3^2,3^3\) $\Rightarrow$ 3 ways
•\(5^1,5^2,5^3\) $\Rightarrow$ 3 ways
•\(7^0,7^1\) $\Rightarrow$ 2 ways
•\(11^0,11^1\) $\Rightarrow$ 2 ways

Step 4: \[ \text{Total} = 3 \times 3 \times 3 \times 2 \times 2 = 108 \]