Download MAT 200, Logic, Language and Proof, Fall 2015 Practice Questions

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.
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