(Smallest common multiple of 12 and 16) `xx` (Smallest common multiple of 10 and 15) is equal to

Options

A. 960
B. 720
C. 1440
D. 480

Answer

To solve the problem of finding the product of the smallest common multiple (LCM) of 12 and 16, and the smallest common multiple of 10 and 15, we will follow these steps:

### Step 1: Find the LCM of 12 and 16
To find the LCM of two numbers, we can use the prime factorization method.

- **Prime factorization of 12**:
- 12 = 2^2 × 3^1
- **Prime factorization of 16**:
- 16 = 2^4

Now, we take the highest power of each prime factor:
- For 2, the highest power is 2^4 (from 16).
- For 3, the highest power is 3^1 (from 12).

Thus, the LCM of 12 and 16 is:
\[ LCM(12, 16) = 2^4 × 3^1 = 16 × 3 = 48 \]

### Step 2: Find the LCM of 10 and 15
Now, we will find the LCM of 10 and 15 using the same method.

- **Prime factorization of 10**:
- 10 = 2^1 × 5^1
- **Prime factorization of 15**:
- 15 = 3^1 × 5^1

Again, we take the highest power of each prime factor:
- For 2, the highest power is 2^1 (from 10).
- For 3, the highest power is 3^1 (from 15).
- For 5, the highest power is 5^1 (from both).

Thus, the LCM of 10 and 15 is:
\[ LCM(10, 15) = 2^1 × 3^1 × 5^1 = 2 × 3 × 5 = 30 \]

### Step 3: Multiply the two LCMs
Now that we have both LCMs, we can multiply them together:
\[ LCM(12, 16) × LCM(10, 15) = 48 × 30 \]

Calculating this gives:
\[ 48 × 30 = 1440 \]

### Final Answer
Thus, the final answer is:
\[ \text{The product of the smallest common multiples is } 1440 \]

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