UCL Online STEP and AEA Preparation Sessions
Session 8: Factors, primes and irrationals
Its prime factorisation reveals a lot of information about a number. In particular it’s a
useful way to understand how many factors a number has, what they are and the
number’s relationship to other numbers via least common factors and highest
common multiples. It will help you to answer questions like:
Find the number of factors of 180. How many of these factors are divisible by 10?
There are also some proof methods such as ‘proof by infinite descent’ which can be
useful in problem-solving and which sometimes depend on demonstrating that
integers have certain factors.
In this session you’ll look at questions involving these ideas as well as discussing
what it means to say that a number is irrational and how irrationality can be dealt
with in mathematical proof and reasoning. Frequently proof by contradiction is used
in mathematical reasoning about irrational numbers - the classic proof that the
square root of 2 is irrational probably the most well-known example of this.
The session will contain a brief discussion of a sketch proof that e is irrational which
was the basis for a STEP question.
Example problem
If a, b, c, d are rational numbers and if  is irrational prove that a  b  c  d  implies
that a  c and b  d .
Being able to use geometrical properties to deduce information about a figure is
important. Solving geometrical problems can also require



good diagrams
experimenting by adding elements such as lines or circles to diagrams to
reveal relationships
using algebra alongside geometry where necessary
In this diagram what is the sum of the four
shaded angles?
This problem is reasonably straightforward
but in producing a solution it would be
important to be clear about which
geometric properties you are using at all
times.
Here is another example
Find the angle between a space diagonal of a cube and
one of its faces. It might not be immediately obvious that
some standard techniques from A level Mathematics will
enable this to be done relatively quickly.
Example problem
Show that the shortest distance from the point  ,   to the line y  mx  c has the value
m  c  
m2  1
.