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Thank you for sharing the image. I see several mathematical problems in this document. I'll analyze and solve them one by one. If you have a specific question you'd like to start with, please let me know. Otherwise, I’ll begin with **Q2 (B, part c)**. Let me know your preference!

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If you're ready, here are the first few problems in detail:

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### **Q2 (B, c): Find the least positive integer \(x\) such that \(13 \mid (x^2 + 1)\)**

This means \(x^2 + 1\) must be divisible by \(13\). Formally, we solve:
\[
x^2 + 1 \equiv 0 \pmod{13}
\]
\[
x^2 \equiv -1 \pmod{13}
\]
\[
x^2 \equiv 12 \pmod{13} \quad \text{(since \(-1 \equiv 12 \mod{13}\))}.
\]

Now, we need to find the values of \(x\) for which \(x^2 \equiv 12 \pmod{13}\). Let’s test the integers from \(x = 0\) to \(x = 12\) (the modular residue classes):

- \(x = 0: 0^2 = 0 \not\equiv 12 \pmod{13}\),
- \(x = 1: 1^2 = 1 \not\equiv 12 \pmod{13}\),
- \(x = 2: 2^2 = 4 \not\equiv 12 \pmod{13}\),
- \(x = 3: 3^2 = 9 \not\equiv 12 \pmod{13}\),
- \(x = 4: 4^2 = 16 \equiv 3 \pmod{13} \not\equiv 12\),
- \(x = 5: 5^2 = 25 \equiv 12 \pmod{13} \quad \text{(solution found)}\).

Thus, the **least positive integer \(x\)** is:
\[
\boxed{5}.
\]

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Would you like me to proceed with the next sub-question or expand on this? Let me know!

**Related Questions:**
1. What is the general solution for quadratic congruences like this one?
2. How can we check for modular inverses efficiently?
3. What is the modular arithmetic property of \((-1)^2\)?
4. How to calculate quadratic residues for arbitrary primes?
5. What if we change \(13\) to another prime?

**Tip:** Always check modular congruences systematically using quadratic residue theory for efficiency.

Find the least positive integer x such that 13 divides x^2 + 1.

Learn how to solve the modular arithmetic problem x^2 + 1 ≡ 0 (mod 13) to find the least positive integer x. Explore quadratic congruences, quadratic residues, and modular arithmetic concepts with a detailed step-by-step solution. Suitable for high school students in Grades 9-12.

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