MAS114 Solutions
Dr James Cranch
Week 2
1. There are many examples: we just give a few possibilities.
(a) Examples include 1 and 123456789;
(b) Examples include 1cfw_2 and 5cfw_7;
?
(c) Examples include , 2, and tanp30q.
2. The sets are:
(a) t0, 1, 2u;
(b
MAS114 Exercises
Dr James Cranch
Week 2
1. Give an example (without proof) of:
(a) a number which is in Z but not in N;
(b) a number which is in Q but not in Z;
(c) a number which is in R but not in Q.
2. Write down all the elements of each of the followi
MAS114 Exercises
Dr James Cranch
Week 1
1. Learn the Greek alphabet: learn the names of all the lower-case letters
, , , , , , , , , , , , , , o, , , , , , , , , .
(Theyre among the commonest of the unfamiliar symbols that mathematicians use.)
2. Work out
MAS114 Exercises
Dr James Cranch
Week 3
1. An even number is an integer that can be written in the form 2k for some
integer k. An odd number is one that can be written in the form 2k ` 1 for
some integer k. Using these denitions, prove the following impli
MAS114 Exercises
Dr James Cranch
Week 4
1. Take any of your solutions from any exercise from Weeks 13 which was
basically correct, but which could benet from being written up more
clearly. Write it up more clearly.
2. Let an be the sum of the rst n positi
MAS114 Solutions
Dr James Cranch
Week 1
1. Youre on your own with this one (but, then, millions
of primary school children in two countries have managed it, so its not so very difcult).
2. The sequences begin with the following seven terms:
(a) 1, 2, 4, 8
MAS114 Solutions
Dr James Cranch
Week 8
1. (a) We can use Euclid to nd that gcdp10, 17q 1. By
working backwards through Euclids algorithm one
can nd a solution to 10x ` 17y 1 and multiply
both sides by 88 to get a solution. However, one
may prefer to spot
MAS114 Exercises
Dr James Cranch
Week 8
1. Find all solutions to the following linear diophantine equations:
(a) 10x ` 17y 88;
(b) 9x ` 15y 100.
2. I went to a shop where widgets cost 36p and gadgets cost 50p, and spent
16.78 on widgets and gadgets. How m
MAS114 Solutions
Dr James Cranch
Week 4
1. Youre on your own here, except that Ive tried to put
readable solutions online for all the exercises so far:
maybe some of those will provide inspiration.
npn`1q
2
n2 pn`1q2
4
2. We saw in lectures that an
for a
MAS114 Solutions
Dr James Cranch
Week 3
1. For the rst part:
(a) If n is even, then n 2k for some integer k. That
means that n2 p2kq2 4k 2 2p2k 2q, which is of
the form 2l. Hence n2 is even.
(b) If n and m are odd, then we can write n 2k ` 1
and m 2l ` 1
MAS114 Solutions
Dr James Cranch
Week 5
1.
(i) To say that a | b is to say that am b for some
integer m. To say that b | c is to say that bn c for
some integer c.
If both of these are true, then we have amn
bn c, and hence a | c, as required.
(ii) This t
MAS114 Exercises
Dr James Cranch
Week 6
1. Compute:
(a) gcdp896, 1200q;
(b) gcdp123456789, 987654321q.
2. Recall that an odd number is one of the form 2k ` 1.
(i) Show that the square of an odd number leaves a remainder of 1 when
divided by 4;
(ii) Show t
MAS114 Exercises
Dr James Cranch
Week 5
1.
(i) Let a, b, c be integers. If a | b and b | c, then show that a | c.
(ii) Let a, b, c be integers. If a | b and a | c, then show that a | b ` c.
(iii) Let a, b, c, d be integers. If a | b and c | d, then show t