The HCF of 65 and 117 is expressible in the form `65m-117`. Find the value of m. also find the LCM of 65 and 117 using prime factorisation method.

Answer

To solve the problem, we need to find the HCF of 65 and 117, express it in the form \(65m - 117\), and then find the value of \(m\). After that, we will find the LCM of 65 and 117 using the prime factorization method.

### Step 1: Finding the HCF of 65 and 117

We will use the Euclidean algorithm to find the HCF.

1. Divide 117 by 65:
\[ 117 = 65 \times 1 + 52 \]
(Remainder is 52)

2. Now, divide 65 by 52:
\[ 65 = 52 \times 1 + 13 \]
(Remainder is 13)

3. Next, divide 52 by 13:
\[ 52 = 13 \times 4 + 0 \]
(Remainder is 0)

Since the last non-zero remainder is 13, the HCF of 65 and 117 is:
\[ \text{HCF} = 13 \]

### Step 2: Expressing HCF in the form \(65m - 117\)

We need to express 13 in the form \(65m - 117\):
\[ 65m - 117 = 13 \]

Rearranging gives:
\[ 65m = 13 + 117 \]
\[ 65m = 130 \]
Now, divide both sides by 65:
\[ m = \frac{130}{65} = 2 \]

### Step 3: Finding the LCM of 65 and 117 using Prime Factorization

1. **Prime factorization of 65**:
\[ 65 = 5 \times 13 \]

2. **Prime factorization of 117**:
\[ 117 = 3 \times 39 = 3 \times 3 \times 13 = 3^2 \times 13 \]

3. **Finding the LCM**:
The LCM is found by taking the highest power of each prime factor:
- From 65: \(5^1\) and \(13^1\)
- From 117: \(3^2\) and \(13^1\)

So, the LCM is:
\[ \text{LCM} = 5^1 \times 3^2 \times 13^1 = 5 \times 9 \times 13 \]

Now calculate:
\[ 5 \times 9 = 45 \]
\[ 45 \times 13 = 585 \]

Thus, the LCM of 65 and 117 is:
\[ \text{LCM} = 585 \]

### Final Answers:
- The value of \(m\) is \(2\).
- The LCM of 65 and 117 is \(585\).

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