Number Theory

When we say "Number Theory" in the context of Maths interviews we are not expecting you to know about modulo arithmetic (though you can of course use it if you want to), or Fermat's little theorem etc.  What we expect you to know are things like:

• Every natural number (greater or equal to 2) can be written uniquely as a product of primes.
• Multiples of three can be written as $3k$, where $k$ is an integer.  Any integer can be written in exactly one of the forms $3k-1$, $3k$ or $3k+1$.  [And you can replace "3" with other integers for similar results.]
• Two consecutive integers can be written as $n$ and $n+1$.

Whilst we don't necessarily expect students to have covered these, a basic understanding of the pigeon-hole principle and arrangements (permutations and combinations) can be useful.

In this video we discuss some Number Theory techniques and some "interview style" questions.  

Interview question examples

Here are some questions for you to try.  Each one has a link to a video where we discuss the question and some possible methods which can be used to solve it.  Make sure you try each question before looking at the video!

Question 1: Is it possible for $2^n-1$ and $2^n+1$ to both be prime numbers? 

Question 2: We have $19=10^2-9^2$, $20=6^2-4^2$ and $21=5^2-2^2$.  Which integers can we write as a difference of two square numbers?

In the video below one of our undergraduates answers questions 1 and 2.

Question 3:  Can we find integers $a, b, c$ and $n$ such that $a^2+b^2+c^2= 8n+7$?

Question 4: Show that the difference between two odd squares is always a multiple of 8.

In the video below one of our undergraduates answers questions 3 and 4.

Question 5: Which integers can be written as a sum of consecutive integers (for example $2+3+4=9$?

This problem appears as "Polite Numbers" on the Nrich website.  There are some proof sorters for different ways to solve it on the page "Impossible Sums".  In the video below one of our undergraduates answers question 5. 

More practice questions

Here are some more questions which involve number theory.  You can also try searching for "number theory" on the STEP database.

• Show that $n^2(n^2-1)(n^2-4)$ is always a multiple of 360 (where $n$ is a positive integer greater than 2).  Here is a video showing a solution.
• Show that all primes greater than 3 are either one more or one less than a multiple of 6.  Here is a video showing a solution.
• "Amicable Arrangements" from NRICH.  This question is based on 2008 STEP 1 Q13, but has been "Christmas-ified".
• "Always Perfect" from NRICH.
• 2010 STEP 1 Q8.
• 1996 STEP 2 Q6.
• 1996 STEP 1 Q3.
• 2013 STEP 2 Q7.
• 2018 STEP 2 Q6.
• 2009 STEP 1 Q1.
• 2014 STEP 1 Q1. 
• 1992 STEP 1 Q7.  
• 1996 STEP 3 Q4. This is a STEP 3 question, so might be quite tricky!
• 2003 STEP 1 Q7.
• Foundation Assignments 6, 10, 12 and 17.