If `M=2^(2)xx3^(5),N=2^(3)xx3^(4)`, then the number of factors of N that are common with factors of M is

Options

A. 8
B. 5
C. 18
D. 15

Answer

To find the number of factors of \( N \) that are common with the factors of \( M \), we first need to determine the prime factorization of both \( M \) and \( N \).

### Step 1: Write the prime factorizations
Given:
- \( M = 2^2 \times 3^5 \)
- \( N = 2^3 \times 3^4 \)

### Step 2: Identify the common prime factors
The common prime factors between \( M \) and \( N \) are \( 2 \) and \( 3 \).

### Step 3: Determine the minimum powers of the common prime factors
For each common prime factor, we take the minimum exponent from \( M \) and \( N \):
- For \( 2 \): The exponent in \( M \) is \( 2 \) and in \( N \) is \( 3 \). The minimum is \( \min(2, 3) = 2 \).
- For \( 3 \): The exponent in \( M \) is \( 5 \) and in \( N \) is \( 4 \). The minimum is \( \min(5, 4) = 4 \).

### Step 4: Write the prime factorization of the common factors
The common factors can be expressed as:
\[ \text{Common Factors} = 2^2 \times 3^4 \]

### Step 5: Calculate the number of factors of the common factors
To find the number of factors of a number given its prime factorization \( p_1^{e_1} \times p_2^{e_2} \), the formula is:
\[ (e_1 + 1)(e_2 + 1) \]
Applying this to our common factors:
- For \( 2^2 \): \( e_1 = 2 \) → \( 2 + 1 = 3 \)
- For \( 3^4 \): \( e_2 = 4 \) → \( 4 + 1 = 5 \)

Now, calculate the total number of common factors:
\[ \text{Number of common factors} = (2 + 1)(4 + 1) = 3 \times 5 = 15 \]

### Final Answer
The number of factors of \( N \) that are common with the factors of \( M \) is \( 15 \).

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