You may use a calculator to help you with simple arithmetic (+,−,×,/)(+,−,×,/)

. No other computational aids are permitted.

Show all your work.on paper with full steps.

Math 4420: Term Test #1

Fall 2020

To be submitted via Crowdmark on Thursday, October 8 within 105 minutes of start time and no

later than 11:45am.

Instructions

1. You may use a calculator to help you with simple arithmetic (+, −, ×, /). No other computational aids are permitted.

2. You must work entirely on your own. You may consult the textbook and your own notes, but

no other sources (online or otherwise) are permitted.

3. Show all your work.

4. You will have 90 minutes to complete the test plus 15 minutes to scan and submit.

Problems

1. Let N be the 3-digit number formed by the 3rd, 4th, and 5th digits of your A#. For example,

A00320517 gives N = 320.

(a) Find the remainder when 3N − 2N is divided by 7.

(b) Find all x ∈ Z such that 293x ≡ 5 (mod N ).

2. (a) Determine all possible values of (2n, n + 6), for n ∈ Z. Explain your reasoning.

(b) Let a, b, c be nonzero integers. Prove that if (a, bc) = 1 and (b, c) = 1 then (ab, c) = 1.

3. (a) Prove there are no nonzero integers a, b such that a4 = 7b2 .

(b) Suppose p is prime and a2 ≡ b2 (mod p). Prove that a ≡ b (mod p) or a ≡ −b (mod p).

4. Consider the formula f ([x]15 ) = [8x]30 .

Show that this yields a well-defined function f : Z15 −→ Z30 and determine whether this

function is injective or surjective. (Justify your answers.)

5. The relation ∼ on Z is defined as follows: a ∼ b if and only if a2 − b2 is a multiple of 5.

Show that ∼ is an equivalence relation and determine all distinct elements of Z/ ∼.

6. This question is for extra credit.

Let p be prime. Prove that p4 + 8 is never prime except in the case p = 3.

1