If HCF of m and n is 1 , then what are the HCF of `m+n , ` and HCF of `m-n` , n respectively ? `( m gt n )`

Options

A. 1 and 2
B. 2 and 1
C. 1 and 1
D. cannot be determined

Answer

To solve the problem step by step, let's analyze the question:

**Question:** If HCF of m and n is 1, then what are the HCF of m+n and m, and the HCF of m-n and n respectively? (Given that m > n)

### Step-by-Step Solution:

1. **Understanding HCF:**
- The HCF (Highest Common Factor) of two numbers is the largest number that divides both of them without leaving a remainder.
- If HCF(m, n) = 1, it means that m and n are coprime (they have no common factors other than 1).

2. **Calculating m + n:**
- Let's denote m + n as a new variable, say P.
- Therefore, P = m + n.

3. **Finding HCF(m + n, m):**
- We need to find HCF(P, m).
- Since P = m + n, we can express it as HCF(m + n, m).
- Using the property of HCF, we know that HCF(a + b, a) = HCF(a, b).
- Therefore, HCF(m + n, m) = HCF(m, n).
- Since HCF(m, n) = 1 (given), we conclude that HCF(m + n, m) = 1.

4. **Calculating m - n:**
- Now, let's denote m - n as another variable, say Q.
- Therefore, Q = m - n.

5. **Finding HCF(m - n, n):**
- We need to find HCF(Q, n).
- Again, using the property of HCF, we have HCF(m - n, n).
- We can express this as HCF(m - n, n) = HCF(m, n) (using the property that HCF(a - b, b) = HCF(a, b)).
- Since HCF(m, n) = 1 (given), we conclude that HCF(m - n, n) = 1.

### Final Conclusion:
- Therefore, the HCF of (m + n, m) is 1.
- The HCF of (m - n, n) is also 1.

### Summary of Results:
- HCF(m + n, m) = 1
- HCF(m - n, n) = 1