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To prove that \( \sqrt{17} \) is irrational, let's proceed with a **proof by contradiction**:

### Step 1: Assume \( \sqrt{17} \) is rational.
If \( \sqrt{17} \) is rational, it can be expressed as a fraction \( \frac{p}{q} \), where:
1. \( p \) and \( q \) are integers,
2. \( \gcd(p, q) = 1 \) (i.e., the fraction is in its simplest form),
3. \( q \neq 0 \).

Thus, we can write:
\[
\sqrt{17} = \frac{p}{q}.
\]

### Step 2: Square both sides.
\[
17 = \frac{p^2}{q^2}.
\]

Multiply through by \( q^2 \) to eliminate the fraction:
\[
17q^2 = p^2.
\]

### Step 3: Analyze the equation.
This equation implies that \( p^2 \) is divisible by \( 17 \). Therefore, \( p \) must also be divisible by \( 17 \) (since \( 17 \) is prime, if it divides \( p^2 \), it must divide \( p \)).

Let \( p = 17k \), where \( k \) is an integer.

### Step 4: Substitute \( p = 17k \) into the equation.
\[
17q^2 = (17k)^2.
\]

Simplify:
\[
17q^2 = 289k^2.
\]

Divide through by \( 17 \):
\[
q^2 = 17k^2.
\]

### Step 5: Analyze the new equation.
This equation implies that \( q^2 \) is divisible by \( 17 \). Therefore, \( q \) must also be divisible by \( 17 \).

### Step 6: Contradiction.
If both \( p \) and \( q \) are divisible by \( 17 \), this contradicts the assumption that \( \frac{p}{q} \) is in its simplest form (\( \gcd(p, q) = 1 \)).

### Conclusion:
The assumption that \( \sqrt{17} \) is rational leads to a contradiction. Therefore, \( \sqrt{17} \) must be irrational.

---

Would you like me to explain a specific step in more detail?  
Here are 5 related questions for further exploration:

1. How can we generalize this proof for any non-perfect square integer \( n \)?
2. What are the properties of prime numbers that make this proof work?
3. Can a similar method be used to prove \( \sqrt{18} \) is irrational?
4. How do we prove that other roots, like \( \sqrt[3]{2} \), are irrational?
5. Is there a geometric interpretation of why \( \sqrt{17} \) is irrational?

**Tip**: When tackling proof questions, breaking the problem into smaller logical steps is key to finding contradictions or verifying claims!

Prove that \( \sqrt{17} \) is irrational.

Understand the step-by-step proof that \( \sqrt{17} \) is irrational using the method of contradiction. This involves number theory concepts, such as rationality and the properties of prime numbers, making it ideal for high school students.

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