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.