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MAS105
SCHOOL OF MATHEMATICS AND STATISTICS
Autumn Semester
2009–2010
Numbers and Proofs
2 hours
Attempt all the questions. The allocation of marks is shown in brackets.
1
(i)
(ii)
(a)
Give the English names for the Greek letters ρ and ζ .
(1 mark)
(b)
Give the symbol for the set of integers.
(1 mark)
Consider the statement:
If a is divisible by 27, then a2 is divisible by 81.
(a)
Write down the converse and the contrapositive of this statement.
(b)
Prove that the original statement is true.
(c)
Is the converse true? Give a proof or a counterexample.
(2 marks)
(3 marks)
(2 marks)
(iii)
What does it mean for a natural number n to be prime? Give the canonical
prime factorisation of 2010.
(4 marks)
(iv)
What is meant by the highest common factor (a, b) of integers a and b, not
both zero?
(2 marks)
(v)
Let a, b, c be integers such that a|bc and (a, b) = 1. Use the existence of
integers s and t satisfying sa + tb = 1 to show that a|c.
(3 marks)
(vi)
(a)
Suppose that n is an odd integer. Show that n2 has remainder 1
when divided by 4.
(2 marks)
(b)
Deduce that no integer of the form 4k + 3 can be written as the sum
of two squares.
(2 marks)
MAS105
1
Turn Over
MAS105
2
(i)
(a)
Find the highest common factor h of 2010 and 1599, and write it in
the form 2010s + 1599t for some integers s and t.
(9 marks)
(b)
Give the general solution to
2010s + 1599t = 111.
What is the smallest non-negative value of s such that this Diophantine equation has a solution?
(7 marks)
(ii)
One of the congruences 1599x ≡ 15 (mod 2010) and 1599x ≡
16 (mod 2010) has no solutions. Which is it? Find all solutions to
the other, expressing your answers in the form x ≡ a (mod m).
(6 marks)
3
4
(i)
(a)
State Fermat's Little Theorem.
(b)
Given that 221 = 13 × 17, compute 32010 (mod 221).
(c)
Dene Euler's φ-function and compute φ(221).
(3 marks)
(10 marks)
(2 marks)
(ii)
Prove, by induction or otherwise, that 4(9)n + 3(2)n is divisible by 7 for all
non-negative integers n.
(7 marks)
(i)
Prove that
(ii)
Give decimal expansions of both a rational number and an irrational number
strictly between 1.2345678901 and 1.2345678912, explaining your answer
briey.
(4 marks)
(iii)
(a)
Explain why the decimal expansion of any rational number m/n
must eventually recur.
(3 marks)
(b)
Write the real number 0.45̇67̇ as the ratio of two coprime natural
numbers.
(6 marks)
(iv)
√
(4 marks)
5 is irrational.
State a simple condition in terms of f and its derivative f 0 for a number
a to be a repeated root of a polynomial f . Hence, or otherwise, show that
the polynomial
x3 + 3x2 + 3x + 3
(5 marks)
has no repeated roots.
End of Question Paper
MAS105
2