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It looks like you've uploaded a midterm examination from a Number Theory course. Would you like help solving specific problems from this exam, or do you want to review the entire document?

Here are a few example problems from the exam:
1. **Problem-solving question:** Find the quotient and remainder when -433 is divided by -17.
2. **LCM and GCD:** Find the greatest common divisor (gcd) and least common multiple (lcm) of specific numbers, such as \(a = 25, b = 40\).
3. **Prime numbers:** Identify pairs of primes satisfying conditions like \(p - q = 3\).
4. **Complex number questions:** Which complex numbers have \(b = 0\)?
5. **Decimal to fraction:** Convert repeating decimals like 0.66... to fractions.

Do you want help with any of these, or another topic from the exam?

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Here are 5 related questions:
1. How do you find the LCM of two numbers using prime factorization?
2. What are twin primes, and can you identify them from a set of numbers?
3. How do you express a repeating decimal as a fraction?
4. How do you calculate the GCD of two numbers using Euclid’s algorithm?
5. How many Mersenne primes are there, and how are they calculated?

**Tip:** For GCD problems, using prime factorizations helps simplify the calculation significantly.

It looks like you've uploaded a midterm examination from a Number Theory course. Would you like help solving specific problems from this exam, or do you want to review the entire document?

Explore detailed solutions to key problems from a Number Theory midterm exam, including GCD, LCM, prime factorization, and prime numbers. Concepts cover university-level mathematics with step-by-step solutions.

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