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MAT 200, Logic, Language and Proof, Fall 2015 Practice Questions Problem 1. Find all solutions to the following linear diophantine equations (i) 165m + 250n = 15; (ii) 26m + 39n = 52; (iii) 33m + 6n = 14. Problem 2. Let a, b be positive integers. Prove that a/(a, b) and b/(a, b) are coprime. Problem 3. Let n ∈ N. Prove that if there are no non-zero integer solutions to the equation xn + y n = z n , then there are no non-zero rational solutions. Problem 4. Find all integers x such that 3x ≡ 15 mod 18. Hint : This is equivalent to solving the linear diophantine equation 3x + 18n = 15. Problem 5. Prove, for positive integers n, that 7 divides 6n + 1 if and only if n is odd. Problem 6. What is the last digit of 32014 ? What about the last digit of 732014 ? Problem 7. Prove that for each positive integer n, there is a sequence of n consecutive integers all of which are composite. Hint : Consider (n + 1)! + 2, (n + 1)! + 3, . . . , (n + 1)! + n + 1. Problem 8. Prove that there are infinitely many prime numbers which are congruent to 3 modulo 4. Hint : Proceed as in the proof of Theorem 23.5.1, but consider m = 4p1 p2 · · · pn − 1. 1