MATH10040: Numbers and Functions
Homework 3: Solutions
1. Let p be a prime number and n > 1. Prove that n p is irrational.
Solution: Suppose FTSOC that n p is rational. Then there exist
nonzero integers a, b satisfying (a, b) = 1 and n p = a/b.
Taking nth
Mechanics of Fluid I Tutorial 6 Turbulent Pipe Flow
Q1.
Describe the mathematical representation of a turbulent flow.
Q2.
Discuss, with diagrams where appropriate, the shear stress and velocity distribution within a pipe.
Q3.
Water at 20C, density 998 kg/
Mechanics of Fluids 1 Lecture 2.1
Pressure Variation in a Fluid
Pressure at a Point
Fluid statics deals with fluids at rest. In a static fluid the pressure
varies due to gravitational field. Pressure is the normal force per
unit area, i.e. the normal stre
UCD School of Mechanical and
Materials Engineering
Reporting / Evaluation Procedures
Laboratory Reports
A laboratory report should typically contain the following sections:
1. TITLE PAGE
Give full details of the name of the laboratory programme (E.g. "Sta
Mechanics of Fluids 1 Lecture 1.1
Fundamental Concepts
Solids, Liquids and Gases
Solids:
densely spaced molecules
large intermolecular cohesive forces
Solids maintain their shape and are not easily deformed!
Mechanics of Fluids 1 Lecture 1.1
Fundamental
UCD School of Electrical, Electronic
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EEEN20020 Laboratory
Design Exercise
Equipment:
DC power supply
Components
Digital Multi-meter
Breadboard
Voltmeter
Oscilloscope
Please Note: Each student must submit a separate individua
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MATH10040
Chapter 4:
Sets, Functions and Counting
1. The language of sets
Informally, a set is any collection of objects. The objects may be mathematical objects such as numbers, functions and even sets, or letters or
symbols of any sort, or objects of th
MATH10040
Chapter 2: Prime and relatively prime
numbers
1. Prime numbers
Recall the basic denition:
Denition 1.1. Recall that a positive integer is said to be prime if it has
precisely two positive divisors (which are necessarily 1 and the number itself )
MATH10040: Numbers and Functions
Homework 1: Solutions
1. P and Q are statements. (Thus each of P and Q is either true or false,
though we may not know which.)
Is the following statement true or false?
Either P implies Q or Q implies P.
Explain your reaso
MATH10040: Numbers and Functions
Homework 4: Solutions
1. Suppose that d, m > 1 and that d|m. Let a, b Z. Prove that if a b
(mod m) then a b (mod d).
Solution: Since d|m, m = dn for some integer n. Suppose that a b
(mod m). Then a b = mt for some t Z. Thu
MATH10040: Numbers and Functions
Homework 2: Solutions
1. Suppose that a, b, c N and that as + bt = c for some s, t Z with
s > 0 and t < 0.
(a) Let k be any integer and let s = s kb, t = t + ka. Show that
as + bt = c.
(b) Prove that there exist S, T Z wit
MATH10040: Numbers and Functions
Homework 5: Solutions
1. How many positive integers less than or equal to 10, 000 are not divisible by 4, 5 or 6?
Solution: Note that a number is divisible by 4 and 5 if and only if it
is divisible by 20 (since (4, 5) = 1)
MATH10040
Chapter 3:
Congruences and the Chinese Remainder
Theorem
1. Congruence modulo m
Recall that Rm (a) denotes the remainder of a on division by m. Thus, by
the division algorithm, 0 Rm (a) < m and a = mt + Rm (a) for some t Z;
The condition a = mt
MATH10040: Numbers and Functions 2013
Mid-term Test
Attempt all questions for full marks. Write your answers in the answer
book provided.
1. (a) Dene what is meant by the greatest common divisor, (a, b), of
two integers a and b.
(b) Prove that if a, m, n
MATH10040: Numbers and Functions
Homework 6: Solutions
1. Prove that
Deduce that
2n
n
2n
n
=2
2n 1
.
n
is always even.
Solution:
2n
(2n)!
=
n
n!n!
2 (2n 1)!
2n 1
2 n (2n 1)!
=
=2
.
=
n
n!n (n 1)!
n!(n 1)!
2n1
n
Since
is an integer, it follows at once that
MATH10040
Chapter 1: Integers and divisibility
1. Divisibilty
Recall the basic denition:
Denition 1.1. If a, b Z, we say that b divides a, or that a is a multiple
of b and we write b|a if there is an integer c such that a = bc.
Example 1.2. Take a = 75, b
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