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Common Divisor

Age 14 to 18
Challenge Level
In this problem you will get the chance to discover some surprising results. Can you prove that they are always true?
Part A
Evaluate $1^3 - 1$, $2^3 - 2$, $3^3 - 3$ and $4^3 - 4$.
What is the largest integer that divides all of these?
Can you prove that it divides $n^3 - n$ for all positive integers $n$?
You should find that your answers are always multiples of 6.
If a number is divisible by 6, which two prime numbers must it be divisible by?
To prove that the expression is always divisible by these two prime numbers, you could start by factorising $n^3 - n$.
Part B
Evaluate $1^5 - 1^3$, $2^5 - 2^3$, $3^5 - 3^3$ and $4^5 - 4^3$.
What is the largest integer that divides all of these?
Can you prove that it divides $n^5 - n^3$ for all positive integers $n$?
Work out which numbers you need to be able to divide by.
Factorising the general expression might be helpful again.
You can consider the cases $n$ even ($n=2k$) and $n$ odd ($n=2k+1$) separately.
Part C
Find the largest integer which divides every member of the following sequence: $$ 1^5-1,\ 2^5-2,\ 3^5-3,\cdots\, n^5-n.$$
Work out which prime numbers you need to be able to divide by. 
Factorise the expression.  Can you show that the expression can be divided by these prime numbers?
You can consider different cases, perhaps starting with $n$ being a multiple of 5 $(n=5k)$.
Part D
Show that $2^{2n} -1$ can always be divided by three.
You can try some values of $n$ and see what happens.
Factorising might again be helpful!
You can consider the expression $2^n$ and what this can be divided by.
Did you know ... ?

Although number theory - the study of the natural numbers - does not typically feature in school curricula it plays a leading role in university at first year and beyond. Having a good grasp of the fundamentals of number theory is useful across all disciplines of mathematics. Moreover, problems in number theory are a great leisure past time as many require only minimal knowledge of mathematical 'content'.

We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.