Number of ways in which a composite number N can be resolved into two factors which are prime to each other if N is of the form `2^2 3^2 5^2 7^2`, is

Options

A. 2
B. `2^(2)`
C. `2^(3)`
D. `2^(4)`

Answer

To find the number of ways in which the composite number \( N = 2^2 \cdot 3^2 \cdot 5^2 \cdot 7^2 \) can be resolved into two factors that are coprime to each other, we can follow these steps:

### Step 1: Understand the Prime Factorization
The number \( N \) can be expressed as:
\[ N = 2^2 \cdot 3^2 \cdot 5^2 \cdot 7^2 \]
Here, the prime factors are \( 2, 3, 5, \) and \( 7 \).

### Step 2: Identify Coprime Factors
Two numbers are coprime if they do not share any prime factors. Therefore, to form coprime factors from \( N \), we can assign each prime factor to either one of the two factors \( A \) or \( B \).

### Step 3: Assign Prime Factors
Since there are 4 distinct prime factors (2, 3, 5, and 7), each prime factor can either go to factor \( A \) or factor \( B \). This gives us \( 2 \) choices (either \( A \) or \( B \)) for each of the \( 4 \) prime factors.

### Step 4: Calculate the Total Combinations
The total number of ways to assign the prime factors is given by:
\[ 2^4 = 16 \]
This includes all possible combinations of assigning the prime factors to either \( A \) or \( B \).

### Step 5: Exclude Invalid Combinations
However, we need to exclude the cases where one of the factors is 1. This occurs when all prime factors are assigned to one factor. There are two such cases:
1. All prime factors go to \( A \) (making \( B = 1 \)).
2. All prime factors go to \( B \) (making \( A = 1 \)).

Thus, we need to subtract these 2 cases from the total:
\[ 16 - 2 = 14 \]

### Step 6: Final Count of Valid Combinations
Therefore, the number of ways in which the composite number \( N \) can be resolved into two coprime factors is:
\[ \boxed{14} \]

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