Match the following.
`{:(" Column I Column II"),("(i) Prime factors of 125 is P. 2 "),("(ii) Common prime factor of 16 and 50 is Q. 4"),("(iii) Smallest composite number is R. 5 "),("(iv) If the L.C.M. of 14 and 21 is 42, then their H.C.F. is S. 7 "):}`

Options

A. `{:("(i) (ii) (iii) (iv) "),("R Q P S"):}`
B. `{:("(i) (ii) (iii) (iv) "),("P R S Q"):}`
C. `{:("(i) (ii) (iii) (iv) "),("R P Q S"):}`
D. `{:("(i) (ii) (iii) (iv) "),("P S Q R"):}`

Answer

To solve the matching question, we will analyze each part of Column I and find the corresponding answer from Column II step by step.

### Step-by-Step Solution:

1. **Finding Prime Factors of 125:**
- To find the prime factors of 125, we start by dividing it by the smallest prime number, which is 5.
- 125 ÷ 5 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1
- Therefore, the prime factorization of 125 is \( 5 \times 5 \times 5 \) or \( 5^3 \).
- The only prime factor is 5.
- **Match:** (i) Prime factors of 125 is **R. 5**.

2. **Finding Common Prime Factor of 16 and 50:**
- First, we find the prime factorization of 16:
- 16 = \( 2 \times 2 \times 2 \times 2 \) or \( 2^4 \).
- Now, we find the prime factorization of 50:
- 50 = \( 5 \times 10 = 5 \times (2 \times 5) = 2 \times 5^2 \).
- The prime factors of 16 are {2} and the prime factors of 50 are {2, 5}.
- The common prime factor is 2.
- **Match:** (ii) Common prime factor of 16 and 50 is **Q. 2**.

3. **Finding the Smallest Composite Number:**
- A composite number is a positive integer that has at least one positive divisor other than one or itself.
- The smallest composite number is 4 (since 1 is neither prime nor composite, and 2 and 3 are prime).
- **Match:** (iii) Smallest composite number is **Q. 4**.

4. **Finding H.C.F. of 14 and 21 given L.C.M.:**
- We know that \( \text{L.C.M.} \times \text{H.C.F.} = \text{Product of the two numbers} \).
- Here, L.C.M. = 42, and the two numbers are 14 and 21.
- Therefore, \( 42 \times \text{H.C.F.} = 14 \times 21 \).
- Calculate \( 14 \times 21 = 294 \).
- Now, \( \text{H.C.F.} = \frac{294}{42} = 7 \).
- **Match:** (iv) If the L.C.M. of 14 and 21 is 42, then their H.C.F. is **S. 7**.

### Final Matches:
- (i) Prime factors of 125 is **R. 5**
- (ii) Common prime factor of 16 and 50 is **Q. 2**
- (iii) Smallest composite number is **Q. 4**
- (iv) If the L.C.M. of 14 and 21 is 42, then their H.C.F. is **S. 7**