Portfolio: Modular arithmetic problems

Portfolio
October 2017
There are four separate handin dates, the instructions for each are on the following pages.
Your work should be submitted on Moodle and each hand in cannot be submitted after
its deadline. There are a total of 60 marks available, but note that not all hand ins
are allocated the same number of marks.
There is a strict limit for each hand in of 1 side of A4. Your submission can be typeset,
or hand-written, but must be submitted as a pdf le.
Marking criteria – The marks are split evenly between
a) mathematical correctness, which includes both the absolute answers and also your
method, and
b) mathematical exposition, where marks are available for structure, quality of explanation,
neatness and clarity.
1
1. Modular arithmetic problems – (10 marks)
You should know your student number
1
, it is 7 digits long. For the purpose of this
section, let N be your student number.
1. Write N in binary, in base 3 and in hexadecimal (with A – F representing 10 – 15).
2. Explain how you can nd N (mod m) for m = 8; 9 and 256, directly from the
answers from the previous question.
3. Find the greatest common divisor of N and 1234567.
4. Find Bezout’s identity for N and 1234567.
2. Set your own problem – (10 marks)
Deadline 13/11/17, 18:00
One particularly important skill as a mathematician is the ability to ask yourself good
questions that challenge your understanding. And attempting to set your own variants
of problems can uncover the true structure of the problem.
1
1. Set a problem at random – Set your own system of linear equations (at least
3 equations in 3 unknowns) by just choosing numbers at random. You don’t need
to show your working, but give the reduced row echelon form, does it have a nice
clean solution?
2. Set a problem systematically – Try to nd a di erent set of linear equations
(again at least 3 equations in 3 unknowns). This time the solution should have a
nice form (no fractions). Explain your methodology.
[Hint: Start with a chosen \nice” solution and then work backwards.]
If for some reason you don’t know your student number then let me know A.S.A.P.
2
3. Hunting to heat exhaustion – (25 marks)
Deadline 27/11/17, 18:00
Primitive hunters were slower and weaker than their larger prey, so caught them not
by running faster, but by overheating them. They would chase away the largest prey
animal they could nd, track it, chase it away again, track it, chase it, etc., each time
not allowing the prey to cool down. Each time the prey would be hotter and so run less
far, until eventually it could run no further through heat exhaustion.
Model
Denote the distance the prey has travelled by D(t), so D(0) = 0. The hunters start at
the same position and run at a constant speed u at all times. Each time the hunters
reach the prey, the prey sets o at a speed v > u and continues on at this speed until it
overheats at which point it stops to recover. The temperature T(t) of the prey at time
t is normalised so that it is initially 0, i.e. T(0) = 0 and so that overheating occurs at
temperature 1. When running the prey heats up at a constant rate h, and when resting
it cools at a constant rate c.
Questions
1. How fast must the hunter run to guarantee that this strategy is successful (i.e. so
that the prey is not permitted to cool down again to 0)?
2. How far will the hunters have to run and how long will it take?
Further notation
Denote the intervals of time when the prey is running by [s
i
; t
], so for the rst run
s
1
= 0. Hence we have
s
<   
You may also want to work with the duration of each run, if so use a
1
< t
1
< s
2
< t
2
< s
3
< t
3
i
and use
b
i
= s
i+1
t
i
for the duration of each recovery interval.
3
i
= t
i
s
i

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