For any integer n, what is HCF (22ra + 7, 33re + 10) equal to?

Options

A. 0
B. 1
C. 11
D. None of these

Answer

To find the HCF (Highest Common Factor) of the expressions \(22n + 7\) and \(33n + 10\) for any integer \(n\), we can follow these steps:

### Step 1: Define the Expressions
We need to find the HCF of the two expressions:
- \(A = 22n + 7\)
- \(B = 33n + 10\)

### Step 2: Substitute Values for \(n\)
To determine the HCF, we can evaluate the expressions for different integer values of \(n\).

#### Case 1: Let \(n = 0\)
- \(A = 22(0) + 7 = 7\)
- \(B = 33(0) + 10 = 10\)

Now, we find the HCF of \(7\) and \(10\):
- The factors of \(7\) are \(1, 7\).
- The factors of \(10\) are \(1, 2, 5, 10\).
- The common factor is \(1\).

Thus, \(HCF(7, 10) = 1\).

### Step 3: Check Another Value for \(n\)
#### Case 2: Let \(n = 1\)
- \(A = 22(1) + 7 = 29\)
- \(B = 33(1) + 10 = 43\)

Now, we find the HCF of \(29\) and \(43\):
- The factors of \(29\) are \(1, 29\) (since \(29\) is prime).
- The factors of \(43\) are \(1, 43\) (since \(43\) is prime).
- The common factor is \(1\).

Thus, \(HCF(29, 43) = 1\).

### Step 4: Check Another Value for \(n\)
#### Case 3: Let \(n = 2\)
- \(A = 22(2) + 7 = 51\)
- \(B = 33(2) + 10 = 76\)

Now, we find the HCF of \(51\) and \(76\):
- The factors of \(51\) are \(1, 3, 17, 51\).
- The factors of \(76\) are \(1, 2, 4, 19, 38, 76\).
- The common factor is \(1\).

Thus, \(HCF(51, 76) = 1\).

### Step 5: General Conclusion
From the evaluations above, we observe that for \(n = 0\), \(n = 1\), and \(n = 2\), the HCF of the two expressions is consistently \(1\).

Thus, we can conclude that for any integer \(n\):
\[ HCF(22n + 7, 33n + 10) = 1 \]

### Final Answer
The HCF of \(22n + 7\) and \(33n + 10\) for any integer \(n\) is \(1\).

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