PRACTISE PROBLEMS FOR SECOND MIDTERM
These are some practise problems for the second midterm in class
on 31st May. The midterm will cover all material covered so far including Chinese Remainder Theore
Chapter 10
Series and Approximations
An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate
Z 1
2
ex dx,
0
you could
Further linear algebra. Chapter I. Integers.
Andrei Yafaev
Number theory is the theory of Z = cfw_0, 1, 2, . . ..
1
Euclids algorithm, B
ezouts identity and
the greatest common divisor.
We say that a
PRACTISE PROBLEMS FOR MIDTERM I
These are some practise problems for the midterm in class on 28th
April 2017. Also look at homework problems assigned so far.
1. (i). Define the greatest common divisor
Practice Midterm Solutions
April 25, 2013
1. (i)Let a and b be nonzero integers. Then gcd(a, b) is the greatest integer d such
that d divides both a and b.
(ii)Let a be odd and b even. One way to show
SELECTED PRACTICE FINAL SOLUTIONS
Solution to Problem 5. Fermats Little Theorem states that if pa, pq 1, then
ap1 1 pmod pq.
piq Note that 149 is prime. We want to show that for every a P Z149 , there
(1) (i). Define the greatest common divisor gcd(a, b) of two nonzero integers a and b.
(ii). Show that gcd(a, b) = gcd(a + b, a + 2b).
(2) State the fundamental theorem of arithmetic. Using it prove
t
v
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)HoASY2dddI)dY)2vdG u !A)w u zAdA)YV
y V ` V ` x G ~ | G V | ` ` G V F GI X y V G ` ` FI ` ` 6 8 4 8 1
HAa92ddd!I dY)2vdG u F AHWdYaHa2