KMB, 18/11/99

M1F Foundations of Analysis—Problem Sheet 7

This is assessed coursework. Please hand in solutions to all questions at the end of the lecture on Friday 26th November.

1^{∗}) In this question, you may assume the (easy) result that if y is a positive real, and m and n are integers, then y^{mn} = (y^{m})^{n}.

a) Prove that if x is a positive real, and m, n are integers, with m > 0, then (using the definitions I gave in the course)

(x^{1/m})^{n} = (x^{n})^{1/m}.

b) Prove that if x is a positive real, and m, n are positive integers, then (x^{1/m})^{1/n} = x^{1/mn}.

c) Prove that if x is a positive real, and s, t are rationals, then (x^{s})^{t} = x^{st}.

2^{∗}) In this question you may assume the standard results about rational powers that we have proved in lectures (and in Q1).

a) Prove that 2^{1/2}5^{1/5}10^{1/10} = (2^{1/5}5^{1/10})^{3}.

b) Prove (\frac{4}{\sqrt{3}})^{−1/2} 12^{3/4} = 3\sqrt{2}.

c) What is the 10th root of 10^{(10^{10})}?

3^{∗}) For both of the following pairs a, b of integers, use the Euclidean algorithm to find the highest common factor d of a and b. Furthermore, find integers λ and μ such that λa + μb = d.

a) a = 42, b = 16.

b) a = 999, b = 185.

4^{∗}) Let a and b be two positive integers. Show, without assuming that integers factor uniquely into primes, that if e is an integer which divides both a and b, then e divides the highest common factor of a and b.

5^{∗}) The highest common factor of 3 and 7 is 1, so there will be integers λ and μ such that 7λ + 3μ = 1. Find an example of such integers λ and μ with λ > 1, 000, 000.