Find the number of ways of expressing 540 as product of two co-prime factors.

Options

A. 7
B. 10
C. 4
D. 6

Answer

To find the number of ways of expressing 540 as a product of two co-prime factors, we can follow these steps:

### Step 1: Prime Factorization of 540
First, we need to find the prime factorization of 540.

\[ 540 = 2^2 \times 3^3 \times 5^1 \]

### Step 2: Identify Co-prime Factors
Two numbers are co-prime if their greatest common divisor (GCD) is 1. To express 540 as a product of two co-prime factors, we can assign the prime factors to two different groups.

### Step 3: Distributing Prime Factors
We can distribute the prime factors among two groups. Each prime factor can either go to the first factor or the second factor.

- The prime factor \(2\) can go to either group.
- The prime factor \(3\) can go to either group.
- The prime factor \(5\) can go to either group.

### Step 4: Calculate the Number of Distributions
Since we have three distinct prime factors, we can assign each of them to one of the two groups (factors). The number of ways to distribute \(n\) distinct items into \(k\) groups is given by \(k^n\).

Here, \(n = 3\) (the prime factors \(2, 3, 5\)) and \(k = 2\) (two groups).

\[ \text{Number of ways} = 2^3 = 8 \]

### Step 5: Adjust for Order
Since the order of the factors does not matter (i.e., \(a \times b\) is the same as \(b \times a\)), we need to divide the total by 2 to avoid double counting.

\[ \text{Number of distinct co-prime pairs} = \frac{8}{2} = 4 \]

### Final Answer
Thus, the number of ways of expressing 540 as a product of two co-prime factors is **4**.

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