If the highest common factor of 65 and 117 is of the form (65m - 117), then $m=$

Question

Accepted Answer

To determine why Option 3 is the correct answer, we need to find the highest common factor (HCF) of 65 and 117 and express it in the form $65m - 117$.

### Step-by-Step Explanation:

1. **Finding the HCF of 65 and 117:**
   - First, we use the Euclidean algorithm to find the HCF.
   - Divide 117 by 65: $117 \div 65 = 1$ remainder $52$.
   - Divide 65 by 52: $65 \div 52 = 1$ remainder $13$.
   - Divide 52 by 13: $52 \div 13 = 4$ remainder $0$.

Since the remainder is now 0, the HCF is the last non-zero remainder, which is 13.

2. **Expressing the HCF in the form $65m - 117$:**
   - We need to find $m$ such that $65m - 117 = 13$.
   - Rearrange the equation: $65m = 13 + 117$.
   - Simplify: $65m = 130$.
   - Solve for $m$: $m = \frac{130}{65} = 2$.

Thus, $m = 2$, which corresponds to Option 3.

### Explanation of Why Other Options are Incorrect:

- **Option 1 (m = 4):**
  - If $m = 4$, then $65m - 117 = 65 \times 4 - 117 = 260 - 117 = 143$.
  - 143 is not the HCF of 65 and 117, as the HCF is 13.

- **Option 2 (m = 3):**
  - If $m = 3$, then $65m - 117 = 65 \times 3 - 117 = 195 - 117 = 78$.
  - 78 is not the HCF of 65 and 117, as the HCF is 13.

- **Option 4 (m = 1):**
  - If $m = 1$, then $65m - 117 = 65 \times 1 - 117 = 65 - 117 = -52$.
  - -52 is not the HCF of 65 and 117, as the HCF is 13.

Therefore, the only value of $m$ that satisfies the condition $65m - 117 = 13$ is $m = 2$, making Option 3 the correct answer.