Divisible Factorisations

Why do this problem?

This problem asks students to use number theory to show that some expressions can always be divided by certain numbers.  There is another version of parts 1 and 2 in the problem Common Divisor, which has more hints within the question and more suggestions for support in the teacher notes.

Possible approach

This article on divisibility tests might be useful, and you could use Dozens as a warm up task.

Students could be asked how they can tell that a number is divisible by $6$.  For example, which of the following must be divisible by $6$ (without actually dividing it!):

$$2574, 8961, 5972, 8981$$

They can also be asked to create a 6 digit number which must be divisible by 6. They can then be asked which numbers we need to be able to divide $n^3 - n$ by to show that it is divisible by 6. Factorising and considering the factors is helpful.

Students could then be asked how they might be able to tell which of the numbers is divisible by $12$.

See also the key ideas section below.

Key questions

Problem 1:  $n^5-n^3$
The second problem draws on the ideas of the first. This time the expressions can be divided by $2$4 so students need to show that it can be divided by $3$ and by $2^3$. The factor of $3$ can be argued from the fact that we have a product of three consecutive numbers, but the factor of $2^3$ is a little trickier. One option is to consider $n$ even and $n$ odd separately.

Problem 2:  $2^{2n}-1$
For the last problem, factorising gives two expressions which differ by $2$. Inserting the in-between number and considering the prime factors of this will help to solve the problem.

Problem 3:  
If $n-1$ is divisible by $3$, then we know that $n-1=3m$ for some integer $m$.

Possible Support

For a version of this problem with more support, and a simpler starting point, given see Common Divisor.

Students could start by working on Take Three from Five where they are introduced to the fact that numbers can be written as a multiple of $3$, $1$ more than a multiple of $3$ or $2$ more than a multiple of $3$.

Students could start by considering $n^2 + n$ and showing that this is always even. The could also show that $n(n+1)(n+2)$ always has a factor of three, possibly by considering $n=3k$, $n=3k+1$ and $n=3k+2$. The fact that in any three consecutive numbers there is always a multiple of three is useful in all of the four problems.

For students who find it difficult to see that any number can be written as one of $3k, 3k+1, 3k+2$ they could consider the three following sequences: $$1, 4, 7, 10, ...$$ $$2, 5, 8, 11, ...$$ $$3, 6, 9, 12, ... $$

Finding the formulae for these three sequences shows that any number has one of the three different forms.

Possible extension

Here is a list of number theory problems.