Question 4. If HCF of $65$ and $117$ is expressible in the form $65m-117$, then the value of $m$ is

Solution:

To solve the problem, we need to find the value of $$m$$ such that the HCF of $$65$$ and $$117$$ can be expressed in the form $$65m-117$$.

1. Find the HCF of 65 and 117:
– The prime factorization of $$65$$ is $$5\times 13$$.
– The prime factorization of $$117$$ is $$3\times 39$$ or $$3\times 3\times 13$$.
– The common factor between $$65$$ and $$117$$ is $$13$$.
– Therefore, $$\text{HCF}(65,117)=13$$.

2. Set up the equation:
– According to the problem, we have $$\text{HCF}=65m-117$$.
– We can substitute the HCF we found into the equation:
$$13=65m-117$$

3. Rearrange the equation to solve for $$m$$:
– Add $$117$$ to both sides:
$$13+117=65m$$
$$130=65m$$

4. Solve for $$m$$:
– Divide both sides by $$65$$:
$$m=\frac{\mathrm{130/}}{65}$$
– Simplifying this gives:
$$m=2$$

Final Answer:
The value of $$m$$ is $$2$$.