9.1 Coursework Sheet 0

Please do not hand in solutions to this worksheet. It is intended to get you thinking at the start of the module.

Exercise 1

Which positive integers have the property that they are divisible by ALL the positive integers from 1 up to half their value?

Exercise 2

Can you prove that every odd integer is a difference of two squares?

Exercise 3

Following on from question 2, which numbers can only be expressed as the difference of two squares in exactly one way? Why?

Exercise 4

Let \(n\) be an odd positive integer. Show that the sum of any \(n\) consecutive integers is divisible by \(n\).

Exercise 5

What is the last digit of \(2^{1,000}\)?

Exercise 6

Let \(n\) be a positive integer greater than \(1\), and \(n!\) denote the product \(1\times 2\times 3\times \ldots \times n\). Show that the integers \(n!+2, n!+3, \ldots n!+n\) are each composite, i.e., each has a factor bigger than \(1\) and less than itself. (This shows that the sequence of prime numbers has arbitrarily long gaps in it.)

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