Find the HCF and LCM of
117, 221

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 117 and 221, we can follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of both numbers.

**For 117:**
- Divide by 3: \(117 \div 3 = 39\)
- Divide 39 by 3: \(39 \div 3 = 13\)
- 13 is a prime number.

So, the prime factorization of 117 is:
\[ 117 = 3^2 \times 13^1 \]

**For 221:**
- Divide by 13: \(221 \div 13 = 17\)
- 17 is a prime number.

So, the prime factorization of 221 is:
\[ 221 = 13^1 \times 17^1 \]

### Step 2: Finding HCF
The HCF is found by taking the lowest power of all common prime factors.

- The common prime factor between 117 and 221 is 13.
- The lowest power of 13 in both factorizations is \(13^1\).

Thus, the HCF is:
\[ \text{HCF} = 13 \]

### Step 3: Finding LCM
The LCM is found by taking the highest power of all prime factors present in either number.

- From 117: \(3^2\) and \(13^1\)
- From 221: \(13^1\) and \(17^1\)

Now, we take the highest powers:
- \(3^2\) from 117
- \(13^1\) from both
- \(17^1\) from 221

Thus, the LCM is:
\[ \text{LCM} = 3^2 \times 13^1 \times 17^1 \]

Calculating this:
\[ \text{LCM} = 9 \times 13 \times 17 \]

Calculating step-by-step:
1. \(9 \times 13 = 117\)
2. \(117 \times 17 = 1989\)

Thus, the LCM is:
\[ \text{LCM} = 1989 \]

### Final Answer
- HCF of 117 and 221 is **13**.
- LCM of 117 and 221 is **1989**.

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