Total number of ways of factorising the number of 676 in to two factors, that are prime to each other is equal to

Options

A. 2
B. 4
C. 1
D. None of these

Answer

To find the total number of ways to factor the number 676 into two factors that are prime to each other, we can follow these steps:

### Step 1: Factor the number 676
First, we need to find the prime factorization of 676.

676 can be expressed as:
\[ 676 = 2^2 \times 13^2 \]

### Step 2: Identify the prime factors
From the prime factorization, we can see that the prime factors of 676 are 2 and 13.

### Step 3: Determine the conditions for factors to be coprime
Two numbers are said to be coprime (or relatively prime) if their greatest common divisor (GCD) is 1. In this case, we want to form two factors from the prime factors 2 and 13 such that they do not share any common prime factors.

### Step 4: Forming the factors
We can form the factors of 676 in the following ways:
1. One factor can be \( 1 \) and the other factor can be \( 676 \) (since \( 1 \) is coprime with any number).
2. The first factor can be \( 2^2 \) (which is 4) and the second factor can be \( 13^2 \) (which is 169).

Thus, we have:
- \( (1, 676) \)
- \( (4, 169) \)

### Step 5: Count the distinct pairs
Now, we need to count the distinct pairs of factors that are coprime. The pairs we found are:
1. \( (1, 676) \)
2. \( (4, 169) \)

These pairs represent the ways to factor 676 into two coprime factors.

### Conclusion
The total number of ways to factor 676 into two coprime factors is **2**.