निम्न में से 12 और 15 का उभयनिष्ठ गुणज कौन है?

Options

A. 1 और 2
B. 1 और 3
C. 3 और 5
D. सिर्फ 3

Answer

To find the least common multiple (LCM) of 12 and 15, we can follow these steps:

### Step 1: Find the prime factorization of each number.
- **For 12:**
- 12 can be divided by 2: \( 12 \div 2 = 6 \)
- 6 can be divided by 2: \( 6 \div 2 = 3 \)
- 3 is a prime number.
- Therefore, the prime factorization of 12 is \( 2^2 \times 3^1 \).

- **For 15:**
- 15 can be divided by 3: \( 15 \div 3 = 5 \)
- 5 is a prime number.
- Therefore, the prime factorization of 15 is \( 3^1 \times 5^1 \).

### Step 2: Identify the highest power of each prime factor.
- From the prime factorization:
- For the prime number 2: The highest power is \( 2^2 \) (from 12).
- For the prime number 3: The highest power is \( 3^1 \) (common in both).
- For the prime number 5: The highest power is \( 5^1 \) (from 15).

### Step 3: Multiply the highest powers of all prime factors.
- Now, we multiply these together to find the LCM:
\[ LCM = 2^2 \times 3^1 \times 5^1 \]
\[ = 4 \times 3 \times 5 \]

### Step 4: Calculate the product.
- First, calculate \( 4 \times 3 = 12 \).
- Then, calculate \( 12 \times 5 = 60 \).

### Conclusion:
- Therefore, the least common multiple (LCM) of 12 and 15 is **60**.

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