Step 1: Understanding the Concept:
To find divisors with specific properties (even and divisible by 15), we first find the prime factorization of the number. Then, we ensure the divisors contain the required factors (at least one 2, at least one 3, and at least one 5).
Step 2: Key Formula or Approach:
If \(N = p_{1}^{a} p_{2}^{b} \dots\), the number of divisors is \((a+1)(b+1)\dots\). For restricted divisors, we adjust the possible ranges for the exponents.
: Detailed Explanation:
1. Prime factorization of \(2079000\):
\[ 2079000 = 2079 \times 10^{3} = (9 \times 231) \times (2 \times 5)^{3} = (3^{2} \times 3 \times 7 \times 11) \times 2^{3} \times 5^{3} \]
\[ 2079000 = 2^{3} \times 3^{3} \times 5^{3} \times 7^{1} \times 11^{1} \]
2. Requirements for the divisor \(d\):
- Even \(\implies\) must contain at least one factor of 2. Exponent of 2 can be \(\{1, 2, 3\}\) (3 choices).
- Divisible by 15 (\(3 \times 5\)) \(\implies\) must contain at least one 3 and at least one 5.
- Exponent of 3 can be \(\{1, 2, 3\}\) (3 choices).
- Exponent of 5 can be \(\{1, 2, 3\}\) (3 choices).
- Exponent of 7 can be \(\{0, 1\}\) (2 choices).
- Exponent of 11 can be \(\{0, 1\}\) (2 choices).
3. Total divisors \(= 3 \times 3 \times 3 \times 2 \times 2 = 108\).
Step 3: Final Answer:
The total number of such divisors is 108.