(Smallest common multiple of 25, 40 and 60) `xx` (Smallest common multiple of 12, 36, 48) is equal to

Options

A. 46800
B. 68400
C. 86400
D. 6848

Answer

To find the solution to the question, we need to calculate the smallest common multiple (LCM) of two sets of numbers: 25, 40, and 60; and 12, 36, and 48. Then we will multiply these two LCMs together.

### Step 1: Find the LCM of 25, 40, and 60

1. **Prime Factorization:**
- 25 = \(5^2\)
- 40 = \(2^3 \times 5\)
- 60 = \(2^2 \times 3 \times 5\)

2. **Identify the highest power of each prime:**
- For \(2\): highest power is \(2^3\) (from 40)
- For \(3\): highest power is \(3^1\) (from 60)
- For \(5\): highest power is \(5^2\) (from 25)

3. **Calculate the LCM:**
\[ \text{LCM} = 2^3 \times 3^1 \times 5^2 = 8 \times 3 \times 25 \]
\[ = 24 \times 25 = 600 \]

### Step 2: Find the LCM of 12, 36, and 48

1. **Prime Factorization:**
- 12 = \(2^2 \times 3\)
- 36 = \(2^2 \times 3^2\)
- 48 = \(2^4 \times 3\)

2. **Identify the highest power of each prime:**
- For \(2\): highest power is \(2^4\) (from 48)
- For \(3\): highest power is \(3^2\) (from 36)

3. **Calculate the LCM:**
\[ \text{LCM} = 2^4 \times 3^2 = 16 \times 9 = 144 \]

### Step 3: Multiply the two LCMs

Now we multiply the two LCMs we found:
\[ 600 \times 144 \]

To calculate:
\[ 600 \times 144 = 86400 \]

### Final Answer
The result is:
\[ \text{The smallest common multiple of 25, 40, 60 and 12, 36, 48 is } 86400. \]

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