In how many ways 12600 can be expressed as a product of two factors which are relatively prime is :

Options

A. a. 12
B. b. 4
C. c. 8
D. d. 72

Answer

To solve the problem of expressing 12600 as a product of two relatively prime factors, we can follow these steps:

### Step 1: Prime Factorization of 12600
First, we need to find the prime factorization of 12600.

- We can break it down as follows:
- 12600 = 126 × 100
- 126 = 2 × 63 = 2 × 3 × 21 = 2 × 3 × 3 × 7 = 2^1 × 3^2 × 7^1
- 100 = 10 × 10 = 10^2 = (2 × 5)^2 = 2^2 × 5^2

Combining these, we get:
- 12600 = 2^3 × 3^2 × 5^2 × 7^1

### Step 2: Identify the Prime Factors
From the prime factorization, we have:
- Prime factors: 2, 3, 5, 7
- Their respective powers: 3, 2, 2, 1

### Step 3: Determine the Combinations of Factors
To express 12600 as a product of two relatively prime factors, we need to assign the prime factors to either of the two factors (let's call them A and B) such that they do not share any prime factors.

- For each prime factor, we have two choices: it can either go to factor A or factor B.

### Step 4: Calculate the Number of Ways
Since we have 4 distinct prime factors (2, 3, 5, 7), and each can independently go to either A or B, we can calculate the total number of combinations as follows:
- Total combinations = 2 (choices for 2) × 2 (choices for 3) × 2 (choices for 5) × 2 (choices for 7) = 2^4 = 16

### Step 5: Adjust for Order
Since the order of the factors does not matter (i.e., A × B is the same as B × A), we need to divide the total combinations by 2:
- Total unique combinations = 16 / 2 = 8

### Final Answer
Thus, the number of ways 12600 can be expressed as a product of two relatively prime factors is **8**.

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