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PAPER TITLE: FURTHER PURE MATHEMATICS
EXAM DATE: MONDAY 8, JUNE 2015
COURSE CODE: M303/J
Question 1
(a) Use the Principle of Mathematical Induction to prove that for all integers n ≥ 2 the number 10n − 4 is divisible by 12.
(b) From the definition of congruence alone, prove the following, for integers a, b, c, d, k, n and r with n > 1 and r > 0.
(i) If a ≡ b (mod n) then ka + c ≡ kb + c (mod n).
(ii) If ka ≡ b (mod n) and kc ≡ d (mod n) then ad ≡ bc (mod n).
(iii) ra ≡ rb (mod rn) if, and only if, a ≡ b (mod n).
(c) Suppose that n ≡ 3 (mod 8). Determine the least positive residue of 5n + 3
(i) modulo 4.
(ii) modulo 20.
ANSWERS(Purchase full paper to get all the solutions):
1a)
is divisible by 12 for all integers n ≥ 2.
For n = 2
For n = 3
Assume n = 2k, since n ≥ 2
(1)
Show that n = 2(k+1). For n ≥ 2
(2)
From (1)
From (2)
(3)
substitute into into (3)
1
12(99D+ 133) is true for all integers n ≥ 2
Hence,
10n − 4 is divisible by 12 for all integers n ≥ 2 is true.
1bi)
Given a ≡ b (mod n)
From the definition of congruence:
a−b = m n
Where m is an integer.
Multiplying both sides by k:
ka−kb = k mn
Since m is an integer, from the definition of congruence, even if k isn’t an integer:
ka≡kb(modkn)
And, since k is an integer then km is also an integer:
Therefore,
ka≡kb(modn)
Add c to both side
ka+c≡kb+c (modn) proved
1bii)
To say that ka is congruent to b modulo n is, by definition, to say that n divides b−ka (without remainder), and that is denoted n|(b−ka).
There are a few properties for divisibility.
The two needed here are
(1) if n divides a number, then it divides every multiple of that number, and
(2) if n divides two numbers, then it divides their subtraction.
Using the above definition of congruence, the question becomes this: if n|(b−ka) and n|(d−kc),
Proof.
Since n|(b−a),
therefore n|(b−a)c, that is, n|(bc−ac). Since K is an integer
Since n|(d−c), , therefore n| a(d−c), that is, n|(ad−ac).
Since n divides both bc−ac and ad−ac, therefore it divides their subtraction,
So, n|(bc−ad).
Hence, If ka ≡ b (mod n) and kc ≡ d (mod n) then ad ≡ bc (mod n). proved
1biii)
ra ≡ rb (mod rn) if, and only if, a ≡ b (mod n).
From definition
multiply both side by r
ra – rb = rmn
ra = rb (mod rn) proved
1ci)
n ≡ 3 (mod 8)
least positive residue of 5n + 3 under modulus 4
n = 11
11 ≡ 3 (mod 8)
5n + 3 = 5(11) +3 = 58
58 mod 4 = 2
the least positive residue of 5n + 3 under modulus 4 = 2
1cii)
least positive residue of 5n + 3 under modulus 20
5n + 3 = 58
58 mod 20 = 18
the least positive residue of 5n + 3 under modulus 20 = 18.
Question 2
Consider the following six groups.
Z12, Z9, Z6 × Z2, Z4 × Z3, Z3 × Z3, Z3 × Z2 × Z2.
(a) (i) List the groups above that are cyclic.
(ii) List the groups above that do not have an element of order 6.
(iii) Write down any pairs of isomorphic groups from the list.
(b) Dic4 is given by the presentation Dic4 = {a, b | a4 = e, a2 = b4, aba3b = e}, with elements expressed in the standard form as arbs for r = 0,..., 3, s = 0,..., 3.
Express the element bab in standard form.
(c) Prove that no group of order 54 is simple.
Question 3
(a) This part of the question is concerned with the σ-function (that is equal to the ‘sum of the positive divisors’). You may assume that this function is multiplicative. Note that if p is prime, then σ(p) = p + 1.
Suppose that a positive integer n has the property that 2n + σ(n) is divisible by 3.
(i) Show that n cannot be prime.
(ii) Show that if n = p2 for some prime p, then p ≡ 2 (mod 3).
(iii) Show that if n = p3 for some prime p, then p ≡ 1 (mod 6).
(b) By considering the field Z7, prove that 8x3 − 6x − 1 is irreducible over Z.
Question 4
Let
A = {(a,b) ∈ R2 :b > 0},
B = {(a,b) ∈ R2: a2 + b2 ≤ 4} and
C = A∩B.
(a) Sketch on separate diagrams, or describe in words, each of A, B and C taking care to indicate which points lie in the sets.
(b) Write down the closure, interior and boundary of C for the following metric.
(i) the Euclidean metric for the plane
(ii) the product metric e on the plane given by
e(( , ,( , )) = | − | + ( , ),
where is the discrete metric for R.
(c) Show that C is not compact for e.
Question 5
This question concerns the ring of Gaussian integers Z[i], with i2 = −1. This ring is a principal ideal domain and a Euclidean domain.
Consider the ideal J of Z[i] given by J = {5, 3 – i}.
(a) By using the Euclidean Algorithm (or otherwise) determine all z ∈ Z[i] such that J = {z}.
(b) Prove that Z[i] /J is a field.
Question 6
A = {f ∈ C[0,1] : |f(x)|≤ x2 for each x, y ∈ [0,1]}
and
B ={f ∈ C[0,1] : ||f(x)−f(y)| ≤ for x, y ∈ [0,1]}
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Last updated: Sep 02, 2021 02:16 PM
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