Question

In how many ways the number 3234 can be written as the product of two numbers which are co-prime to each other?

• A. \(7 \)
• B. \(9 \)
• C. \(8 \)
• D. \(10 \)
• E. \(11 \)

Hint:

For co-prime factor pairs: - Count distinct prime factors $k$ - Number of ways = $2^{k-1}$

Answer 1

The Correct Option is C

Concept: If a number is expressed as: \[ N = p_1^{a_1} \cdot p_2^{a_2} \cdots p_k^{a_k} \] Then the number of ways to write $N$ as a product of two co-prime numbers is: \[ 2^{k-1} \] where $k$ is the number of distinct prime factors.
Step 1: Prime factorization of 3234.
\[ 3234 = 2 \times 3 \times 7^2 \times 11 \] Distinct prime factors: \[ 2, 3, 7, 11 \Rightarrow k = 4 \]

Step 2: Apply formula.
\[ \text{Number of ways} = 2^{k-1} = 2^{4-1} = 2^3 = 8 \]