What will be the least number which when doubled, will be exactly divisible by 4, 6, 9, 12 and 14?

Options

A. 126
B. 252
C. 504
D. 63

Answer

To find the least number which when doubled is exactly divisible by 4, 6, 9, 12, and 14, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
To solve this problem, we first need to find the Least Common Multiple (LCM) of the numbers 4, 6, 9, 12, and 14. The LCM is the smallest number that is a multiple of all the given numbers.

### Step 2: Prime Factorization
We will perform prime factorization for each of the numbers:
- **4** = \(2^2\)
- **6** = \(2^1 \times 3^1\)
- **9** = \(3^2\)
- **12** = \(2^2 \times 3^1\)
- **14** = \(2^1 \times 7^1\)

### Step 3: Determine the Highest Powers of Each Prime
Next, we take the highest power of each prime number from the factorizations:
- For \(2\): The highest power is \(2^2\) (from 4 and 12).
- For \(3\): The highest power is \(3^2\) (from 9).
- For \(7\): The highest power is \(7^1\) (from 14).

### Step 4: Calculate the LCM
Now we can calculate the LCM using these highest powers:
\[ \text{LCM} = 2^2 \times 3^2 \times 7^1 \]
Calculating this gives:
\[ \text{LCM} = 4 \times 9 \times 7 \]
Calculating step-by-step:
- \(4 \times 9 = 36\)
- \(36 \times 7 = 252\)

So, the LCM of 4, 6, 9, 12, and 14 is 252.

### Step 5: Find the Required Number
Since we need the least number which when doubled is divisible by the LCM, we set up the equation:
\[ 2x = 252 \implies x = \frac{252}{2} = 126 \]

### Conclusion
The least number which when doubled will be exactly divisible by 4, 6, 9, 12, and 14 is **126**.

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