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MA3Z7 2023/24 A 800
MA3Z7 2022/23 A 850
NUMBER THEORY (MA3Z7)
April/May 2024
SECTION A
1. (a) Use the Euclidean algorithm to find d = (936, 2200). [7 marks]
(b) Use your solution to part (a) to findh, k ∈ Z satisfying
d = 936h + 2200k. [10 marks]
(c) When we write a natural number n in base-6, we write n as a string nk . . . n0 of digits ni ∈ {0, 1, 2, 3, 4, 5} uniquely determined so that
n = nk 6k + nk -16k-1 + . . . + n1 6 + n0 .
Show that n is divisible by 5 if and only if
5 j nk + nk -1 + . . . + n1 + n0 . [5 marks]
2. In this question, you must justify all of your assertions, but you may refer to results from the lecture notes without proof.
(a) Find all solutions to the linear congruence
14x 三 21 (mod 63)
by reducing it to a linear congruence with a unique solution (modulo m). [7 marks]
(b) Find all solutions to the following system of congruences.
2x 三 3 (mod 5) 3x 三 5 (mod 7) 5x 三 7 (mod 11) [12 marks]
(c) Show that the congruence
x2 三 3 (mod 5005)
has no solutions. [4 marks]
3. In this question, you may use any results from the lecture notes without proof.
(a) Find the remainder when we divide 3621 by 8. [9 marks]
(b) Find the exponent of 3 in the prime factorisation of 2024!. [6 marks]
SECTION B
Throughout, you must justify all of your answers.
4.
(a)
(i) Give an example of an arithmetic function that is not multiplicative. [3 marks]
(ii) Give an example of an arithmetic function that is multiplicative but not completely multiplicative. You need not prove that your function is multiplicative if this was established in lectures. [3 marks]
(iii) Find the sum of the divisors of 385. You may assume any formula given in lectures. [4 marks]
(b) Express the generating function of the divisor function in terms of the Riemann zeta function. You may use any results from the lecture notes as long as you state them precisely. [9 marks]
(c) Let λ(n) = (—1)k where k is the total number of prime factors of n (so λ(12) = λ(2 · 2 · 3) = (—1)3 = —1).
(i) Show that the generating function of λ(n) is ζ(2s)ζ(s)—1 . You may assume that λ(n) is completely multiplicative. [7 marks]
(ii) Show that
5. (a) State and prove the fundamental property of the Von Mangoldt function Λ(n). [11 marks]
(b) Express the generating function of σ(n) in terms of the Riemann zeta function. You may use any results from the lecture notes as long as you state them precisely. [9 marks]
(c) Show that
You may use any results from the lecture notes without proof. [20 marks]