What is the smallest positive integer n, where n \ne 11, such that the highest common factor of n – 11 and 3n + 20 is greater than 1?

64

60

56

59

To solve this problem, we need to find the smallest positive integer \( n \) (where \( n \ne 11 \)) such that the highest common factor (HCF) of \( n - 11 \) and \( 3n + 20 \) is greater than 1.

Let's define:
\[
a = n - 11, \quad b = 3n + 20
\]

We need to find \( n \) such that \( \gcd(a, b) > 1 \).

### Step-by-Step Approach:

1. **Express \( b \) in terms of \( a \):**

   Since \( a = n - 11 \), we have:
   \[
   n = a + 11
   \]

   Substitute \( n = a + 11 \) into \( b = 3n + 20 \):
   \[
   b = 3(a + 11) + 20
   \]
   Simplify this:
   \[
   b = 3a + 33 + 20 = 3a + 53
   \]

2. **Finding \( \gcd(a, b) \):**

   Now, we need to find \( \gcd(a, 3a + 53) \). Using the Euclidean algorithm:

   \[
   \gcd(a, 3a + 53) = \gcd(a, (3a + 53) \mod a)
   \]

   Since \( (3a + 53) \mod a = 53 \mod a \), we get:
   \[
   \gcd(a, 3a + 53) = \gcd(a, 53)
   \]

   For the HCF to be greater than 1, \( a \) must be a divisor of 53. The divisors of 53 are 1 and 53 (since 53 is a prime number).

   We need \( \gcd(a, 53) > 1 \), so \( a = 53 \).

3. **Determine the corresponding \( n \):**

   Recall that \( a = n - 11 \), so:
   \[
   53 = n - 11 \implies n = 64
   \]

   Since \( n \ne 11 \), and we are looking for the smallest positive integer \( n \), the answer is:

\[
\boxed{64}
\]

### Verification:

- When \( n = 64 \):
  - \( n - 11 = 64 - 11 = 53 \)
  - \( 3n + 20 = 3 \cdot 64 + 20 = 192 + 20 = 212 \)
  - \( \gcd(53, 212) = 53 \), which is greater than 1.

Thus, the smallest positive integer \( n \) satisfying the conditions is indeed \( \boxed{64} \). 

### Follow-Up Questions:

1. What is the Euclidean algorithm, and how is it applied to finding the GCD?
2. Why does the divisor of 53 determine \( a \), and why must it be greater than 1?
3. How do we handle cases where both expressions involve variables in finding their GCD?
4. What are some properties of GCDs when one of the numbers is prime?
5. Could there be other integers \( n \) that satisfy the condition for different expressions?

**Tip:** When solving GCD problems involving expressions, try simplifying both expressions in terms of one variable and use the Euclidean algorithm effectively.

Discover the smallest positive integer n (n ≠ 11) where the greatest common divisor (GCD) of n - 11 and 3n + 20 is greater than 1. Follow a detailed step-by-step solution using number theory principles and the Euclidean algorithm.

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