MATH 311 Practice Midterm

October 2014

Queen’s University Department of Mathematics

Faculty of Arts and Science Dr. Thomas A. Hulse

Instructions: This is not homework. This is a practice exam. It is meant to giveyou a sense of what the written midterm for MATH 311 will be like. While you areallowed to use this test however you like, I strongly encourage you try this under closed-book and timed conditions after having first reviewed and absorbed the material. Ifany questions surprise you or are particularly challenging, you may want to go back andreview the associated material. DO NOT USE THIS AS A STUDY GUIDE, it is notcomprehensive and questions on the actual exam may be completely different.

The exam has ten questions labelled 1 through 6. Each question is worth 10 marks.

The exam is meant to take two hours.

In the real test, to receive full credit you must explain your answers, unless otherwisestated. I recommend you try doing that for practice purposes.

Write all answers on the exam. You may use the backs of pages if necessary.

Queen’s approved calculators are permitted, as they will be permitted on the actualexam. I don’t think they’re needed.

Solutions are available on the course website, as are other preparation materials.

Good luck.

Student Number:

This material is copyrighted and is for the sole use of students registered in MATH/MTHE326 and writing this examination. This material shall not be distributed or dissemi-nated. Failure to abide by these conditions is a breach of copyright and may constitutea breach of academic integrity under the University Senate’s Academic Integrity PolicyStatement.

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Page 1 of 7

MATH 311 October 2014 Page 2 of 7

1. — Let n ∈ N.

a) [3 pts] Show that if p ≡ 1 (mod 4) for every prime p|n then n ≡ 1 (mod 4).

b) [3 pts] By part (a), show that if n ≡ 3 (mod 4) then p ̸≡ 0 or 2 (mod 4) for everyprime p|n and p ≡ 3 (mod 4) for at least one prime p|n. You may use part (a).

c) [4 pts] Use part (b) to prove that there are an infinite number of primes p such thatp ≡ 3 (mod 4). [Hint: Consider the divisors of 4N − 1 when N ∈ N.]

MATH 311 October 2014 Page 3 of 7

2. — Let n,m ∈ N such that (n,m) = 1.

a) [2 pts] Show that mn− 1 ≡ m− 1 + n− 1 (mod 2).

b) [2 pts] Let f : N → R where f(n) = (−1)n−1. Use part (a) to show that f ismultiplicative.

c) [6 pts] Let

g(n) =!

d|n

µ(n)f(n).

Compute g(pj) where p is an odd prime and j ∈ N. Compute g(2k) where k ∈ N.Deduce that for n ∈ N with n ≥ 2

g(n) =

"

2 if n = 2k for k ∈ N

0 otherwise.

Show your work.

MATH 311 October 2014 Page 4 of 7

3. —

a) [5 pts] Note that 561 = 3 · 11 · 17 and prove that for any a ∈ Z such that (a, 561) = 1we have a560 ≡ 1 (mod 561).

b) [5 pts] Show that if m,n ∈ N such that (m,n) = 1 then

mφ(n) + nφ(m) ≡ 1 (mod mn).

MATH 311 October 2014 Page 5 of 7

4. —

a) [5 pts] Show that if p is an odd prime and p+ 2 is also a prime then p is a quadraticresidue (mod (p+ 2)) iff p ≡ ±1 (mod 8).

b) [5 pts] The number 709 is prime. Determine if 103 is a quadratic residue (mod 709).Show your work.

MATH 311 October 2014 Page 6 of 7

5. —

a) [3 pts] Find all primitive roots (mod 7).

b) [7 pts] Find all solutions x, y ∈ Z to

y2 ≡ 3x3 (mod 7).

MATH 311 October 2014 Page 7 of 7

6. —

a) [6 pts] If n = qp+1 where q and p are both primes, show that the order of q (modn) is 2q.

b) [4 pts] Show that if n = qp + 1 then p|φ(n).