Find the HCF and the LCM for the following sets of numbers
16,20,48

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 16, 20, and 48, we will follow these steps:

### Step 1: Prime Factorization
First, we will find the prime factorization of each number.

- **For 16**:
- 16 can be divided by 2:
- 16 ÷ 2 = 8
- 8 ÷ 2 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1
- Therefore, the prime factorization of 16 is:
\[ 16 = 2^4 \]

- **For 20**:
- 20 can be divided by 2:
- 20 ÷ 2 = 10
- 10 ÷ 2 = 5
- 5 ÷ 5 = 1
- Therefore, the prime factorization of 20 is:
\[ 20 = 2^2 \times 5^1 \]

- **For 48**:
- 48 can be divided by 2:
- 48 ÷ 2 = 24
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 ÷ 3 = 1
- Therefore, the prime factorization of 48 is:
\[ 48 = 2^4 \times 3^1 \]

### Step 2: Finding HCF
To find the HCF, we take the lowest power of all common prime factors.

- The common prime factor is 2.
- The lowest power of 2 among the factorizations:
- From 16: \(2^4\)
- From 20: \(2^2\)
- From 48: \(2^4\)

Thus, the lowest power is \(2^2\).

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

### Step 3: Finding LCM
To find the LCM, we take the highest power of all prime factors present in any of the numbers.

- For 2: The highest power is \(2^4\) (from both 16 and 48).
- For 5: The highest power is \(5^1\) (from 20).
- For 3: The highest power is \(3^1\) (from 48).

Now, we multiply these together to find the LCM:
\[ LCM = 2^4 \times 5^1 \times 3^1 \]
Calculating this step by step:
- First, calculate \(2^4 = 16\).
- Then, multiply by 5: \(16 \times 5 = 80\).
- Finally, multiply by 3: \(80 \times 3 = 240\).

Therefore, the LCM is:
\[ LCM = 240 \]

### Final Answer
- HCF = 4
- LCM = 240

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