Find the HCF of 144, 336 and 2016 ?

Answer

To find the HCF (Highest Common Factor) of the numbers 144, 336, and 2016, we will follow these steps:

### Step 1: Prime Factorization of Each Number
We need to express each number as a product of its prime factors.

1. **For 144**:
- 144 can be factored as \( 2 \times 2 \times 2 \times 2 \times 3 \times 3 \).
- This can be written in exponential form as \( 2^4 \times 3^2 \).

2. **For 336**:
- 336 can be factored as \( 2 \times 2 \times 2 \times 2 \times 3 \times 7 \).
- This can be written in exponential form as \( 2^4 \times 3^1 \times 7^1 \).

3. **For 2016**:
- 2016 can be factored as \( 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 \).
- This can be written in exponential form as \( 2^5 \times 3^2 \times 7^1 \).

### Step 2: Identify the Common Prime Factors
Now we will identify the common prime factors from the factorizations:

- **For the prime factor 2**:
- The powers are \( 4 \) (from 144), \( 4 \) (from 336), and \( 5 \) (from 2016).
- The minimum power is \( 4 \).

- **For the prime factor 3**:
- The powers are \( 2 \) (from 144), \( 1 \) (from 336), and \( 2 \) (from 2016).
- The minimum power is \( 1 \).

- **For the prime factor 7**:
- The powers are \( 0 \) (from 144), \( 1 \) (from 336), and \( 1 \) (from 2016).
- The minimum power is \( 0 \).

### Step 3: Calculate the HCF
Now we can calculate the HCF using the lowest powers of the common prime factors:

\[ \text{HCF} = 2^{\text{min}(4)} \times 3^{\text{min}(1)} \times 7^{\text{min}(0)} \]

This simplifies to:

\[ \text{HCF} = 2^4 \times 3^1 \times 7^0 = 16 \times 3 \times 1 = 48 \]

### Final Answer
Thus, the HCF of 144, 336, and 2016 is **48**.

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