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 inclu
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
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 u
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 inte
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
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 sequenc
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 b
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
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.
np
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
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 ha
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
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