How many times is the HCF of 32, 36, 72 and 24 contained in their LCM?

Options

A. 16
B. 64
C. 72
D. 84

Answer

To solve the problem of how many times the HCF (Highest Common Factor) of the numbers 32, 36, 72, and 24 is contained in their LCM (Lowest Common Multiple), we will follow these steps:

### Step 1: Find the HCF of the numbers 32, 36, 72, and 24.

To find the HCF, we can use the prime factorization method:

- **32** = \(2^5\)
- **36** = \(2^2 \times 3^2\)
- **72** = \(2^3 \times 3^2\)
- **24** = \(2^3 \times 3^1\)

Now, we take the lowest power of each prime factor that appears in all the numbers:

- For \(2\): The lowest power is \(2^2\) (from 36).
- For \(3\): The lowest power is \(3^0\) (since 32 has no factor of 3).

Thus, the HCF is:
\[ HCF = 2^2 = 4 \]

### Step 2: Find the LCM of the numbers 32, 36, 72, and 24.

To find the LCM, we take the highest power of each prime factor that appears in any of the numbers:

- For \(2\): The highest power is \(2^5\) (from 32).
- For \(3\): The highest power is \(3^2\) (from 36 and 72).

Thus, the LCM is:
\[ LCM = 2^5 \times 3^2 \]

Calculating this gives:
\[ LCM = 32 \times 9 = 288 \]

### Step 3: Determine how many times the HCF is contained in the LCM.

Now we need to find how many times the HCF (which is 4) is contained in the LCM (which is 288):
\[ \text{Number of times} = \frac{LCM}{HCF} = \frac{288}{4} = 72 \]

### Final Answer:
The HCF of 32, 36, 72, and 24 is contained in their LCM **72 times**.

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