MATH0034 Exam 2022 Guidelines and Sample Questions
C. Busuioc
1 Format
The exam consists of 4 questions, each work 25 marks.
Each question has 3 or 4 short subquestions, not necessarily related.
Half of the exam consists of computational exercises, similar to the ones you encountered in the
STACK quizzes during the term. In the computational exercises, you should explain your steps- the right
final answer without any further explanations would not be awarded any marks.
The other half consists of questions that are more theoretical in nature, either slight variations or
special cases of results proven in the lectures or similar to the ones encountered in the problem sheets.
You may use any results proved in lectures, problem classes or which appeared on the problem sheets
without proof, but you have to state them clearly for full marks.
2 Timing
The exam takes place online, on Crowdmark, as a timed 2-hour exam +20 minutes for upload (these are
the current UCL guidelines).
3 Revision Advice
Go through the lectures and problem class notes carefully, do all the questions on the problem sheets and
the practice quizzes and try to solve as many exercises as you can from the past exam papers and the
textbook or the other references for the module. With a solid revision of all the material covered in the
module and a lot of practice, you will gain the skills to do well on the exam.
4 Sample Exam Questions
1. (a) (6 marks) Solve the congruence
x955 ≡ 11 mod 3590
(b) (8 marks) Let p and q be distinct odd prime numbers. Prove that for any integer x,
x(pq−p−q+3)/2 ≡ x mod pq.
(c) (5 marks)
Which of the following are primitive roots modulo the prime 227?
1, 2, 39.
(d) (6 marks) For which prime numbers p ∈ N does the congruence x2 ≡ −5 mod p have solutions?
Your answer should be given in terms of congruences mod 20.
Solution. See problem set 1, exercises 5 and 8 and problem set 2, exercises 5 and 7.
2. (a) (5 marks) Determine the Legendre symbol
(
123
179
)
.
(b) (7 marks) Show that if a prime p ∈ N can be written as 2a2 − 2ab+ 3b2 for integers a and b,
then p cannot be congruent to 11 mod 20.
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(c) (6 marks) Show that for any odd prime p, the Gauss, G(p), can be rewritten as
G(p) =
p−1∑
x=0
ζx
2
p .
(d) (7 marks) Let p ∈ N be a prime number, p ≡ 3 mod 4, p > 3. Suppose q = 2p + 1 is also
prime. Show that 2p − 1 is not prime.
Solution. For (b), (c) and (d) see problem set 2 exercises 4 and 6 and problem set 3 exercise 2.
3. (a) (6 marks) Let f(x) = x7 + 200x4 + x3 + 9x+ 32. Remark that f(1) ≡ 0 mod 3.
Find a solution to the congruence
f(x) ≡ 0 mod 36.
(b) i. (6 marks) Using the fact that log(1 + 3x) converges 3-adically for all x ∈ Z(3), solve the
congruence
28x ≡ 55 mod 37.
ii. (5 marks) Solve the congruence
x28 ≡ 55 mod 37.
(c) (8 marks) Let
αn = 2 +
22
2
+
23
3
+ ....+
2n
n
.
Show that the 2-adic valuation of αn, v2(αn), satisfies the inequality
v2(αn) ≥ min
t≥n+1
(t− v2(t)).
Solution. See Exam 2021, questions 3 and 6.
4. (a) i. (5 marks) Determine the value of the continued fraction [5; 5, 10].
ii. (5 marks) Find two solutions (x, y) in positive integers to Pell’s equation
x2 − 27y2 = 1.
(b) (6 marks)Find a solution x, y ∈ Z to the diophantine equation
x2 + yx+ y2 = 403.
(c) (9 marks) Show that the value of the convergent series
∞∑
k=1
1
23k
is an irrational number.
Solution. See Exam 2021, questions 4 and 6.
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