1. GCSE: Counterexamples & Proofs Involving Integers
Dr J
Last modified: 4th October 2019

2. www.drfrostmaths.com Everything is completely free. Why not register?
Register now to interactively practise questions on this topic, including past paper questions and extension questions (including UKMT).
Teachers: you can create student accounts (or students can register themselves), to set work, monitor progress and even create worksheets.
With questions by:
Dashboard with points, trophies, notifications and student progress.
Questions organised by topic, difficulty and past paper.
Teaching videos with topic tests to check understanding.

3. Bad examples of proof
“Show that the sum of any three consecutive integers is a multiple of 3.”
Kyle’s proof:
“ = 18, which is divisible by 3”.
Que Problemo?
He hasn’t shown it’s true for all possible integers.
Given this example, it might be true, but it might not work for other examples. ?
Key Terms: An integer just means a whole number.

4. How Many Examples Needed?
In 1772 Euler noticed that the following equation gives prime numbers for many positive integers n:
𝒏𝟐 –𝒏+𝟒𝟏
We might wonder if the statement “For all positive integers n, n2 – n + 41 is prime” is true.
Try this for a few 𝒏. How many examples of 𝒏 would we need to show this statement is:
True
False
One
A Few
Infinitely Many
One
A Few
Infinitely Many
To prove a statement is always true, trying a few examples won’t be enough. After trying hundreds of examples, we might be fairly sure something is true, but we wouldn’t be absolutely sure.
The smallest value of 𝑛 for which this statement is false is: 41 ?

5. Counterexamples
! A counterexample is an example used to disprove a statement.
Discussing in pairs, find counterexamples for the following statements:
Statement
Possible counterexample
Prime numbers are always odd.
2 is prime, but not odd.
The square root of a number is always smaller than the number itself.
0.25 =0.5, but 0.5>0.25.
If 𝑝 is prime then 𝑝+2 is prime.
7 is prime but 9 is not.
2 𝑛 is prime for all integer values of 𝑛
(2× 11 2 )+11 will be divisible by 11 because 2× and 11 both are. 1 ? 2 ? 3 ? 4 ?

6. “Show that for any integer 𝑛, 𝑛 2 +𝑛 is always even.”
Even/Odd Proofs
The key here is that all integers are either odd or even.
We can therefore do a ‘case analysis’ and consider these two different cases.
If you later do A Level Maths, you will see this is called a proof by exhaustion, as we’re considering all possible cases.
“Show that for any integer 𝑛,
𝑛 2 +𝑛 is always even.”
If 𝑛 is odd: 𝑜𝑑𝑑×𝑜𝑑𝑑+𝑜𝑑𝑑 = 𝑜𝑑𝑑+𝑜𝑑𝑑 =𝑒𝑣𝑒𝑛
If 𝑛 is even: 𝑒𝑣𝑒𝑛×𝑒𝑣𝑒𝑛+𝑒𝑣𝑒𝑛 = 𝑒𝑣𝑒𝑛+𝑒𝑣𝑒𝑛 =𝑒𝑣𝑒𝑛
Therefore 𝑛 2 +𝑛 is even for all integers. ?

7. Test Your Understanding
Remember, consider the two cases:
When 𝑛 is even…
When 𝑛 is odd…
Prove that 𝑛 𝑛−1 +1 is odd for all integers 𝑛.
If 𝒏 is odd, 𝒐𝒅𝒅×𝒆𝒗𝒆𝒏+𝒐𝒅𝒅=𝒐𝒅𝒅
If 𝒏 is even, 𝒆𝒗𝒆𝒏×𝒐𝒅𝒅+𝒐𝒅𝒅=𝒐𝒅𝒅
Thus 𝒏 𝒏−𝟏 +𝟏 is odd for all integers 𝒏. A ?
[IMC 2009 Q16] How many different positive integers 𝑛 are there for which 𝑛 and 𝑛 2 +3 are both prime numbers? A 0 B 1 C D 3 E infinitely many Solution: B All prime numbers except 2 are odd. But 𝒐𝒅 𝒅 𝟐 +𝒐𝒅𝒅=𝒆𝒗𝒆𝒏, which can’t be prime (and note 𝒏 𝟐 +𝟑 can’t be 2). This the only possibility is if 𝒏=𝟐.
Indeed 𝟐 𝟐 +𝟑=𝟕 is prime. So there is one solution only. N
(Hint: Think odd/even) ?

8. Exercise 1
[SMC 2001 Q9] Which of the following numbers 𝑛 gives a counter-example for the statement: “If 𝑛 is a prime number then 𝑛 2 +2 is also a prime number’? A 3 B 5 C 6 D 9 E none of them Solution: B
[SMC 2006 Q12] The factorial of 𝑛, written 𝑛! Is defined by 𝑛!=1×2×3×…×𝑛. Which of the following values of 𝑛 provides a counterexample to the statement: “If 𝑛 is a prime number, then 𝑛!+1 is also a prime number”? A 1 B 2 C D 4 E 5 Solution: E
[SMC 2013 Q16] Andrew states that every composite number of the form 8𝑛+3, where 𝑛 is an integer, has a prime factor of the same form. Which of these numbers is an example showing that Andrew’s statement is false? A 19 B 33 C D 91 E 99 Solution: D 6 1
Prove that 2𝑛+1 is odd for all integers 𝑛.
If 𝒏 is odd, 𝒆𝒗𝒆𝒏×𝒐𝒅𝒅+𝒐𝒅𝒅=𝒐𝒅𝒅
If 𝒏 is even, 𝒆𝒗𝒆𝒏×𝒆𝒗𝒆𝒏+𝒐𝒅𝒅=𝒐𝒅𝒅
Therefore 𝟐𝒏+𝟏 is odd for all integers 𝒏.
Prove that 𝑛 3 −𝑛 is always even for all integers 𝑛.
If 𝒏 is odd, 𝒐𝒅𝒅×𝒐𝒅𝒅×𝒐𝒅𝒅−𝒐𝒅𝒅=𝒆𝒗𝒆𝒏
If 𝒏 is even, 𝒆𝒗𝒆𝒏×𝒆𝒗𝒆𝒏×𝒆𝒗𝒆𝒏−𝒆𝒗𝒆𝒏=𝒆𝒗𝒆𝒏
Thus 𝒏 𝟑 −𝒏 is always even for all integers 𝒏.
Find the smallest positive counterexample for the statement “If 𝑝 is prime, then 𝑝 2 +𝑝+1 is prime.”
Smallest is 𝒑=𝟕 as 𝟕 𝟐 +𝟕+𝟏=𝟓𝟕=𝟏𝟗×𝟑
[Kangaroo Pink 2015 Q16] Which of the following values of 𝑛 is a counterexample to the statement, ‘If 𝑛 is a prime number, then exactly one of 𝑛−2 and 𝑛+2 is prime’? A 11 B 19 C D 29 E 37 Solution: E
[SMC 2013 Q15] For how many positive integers 𝑛 is 4 𝑛 −1 a prime number? A 0 B 1 C 2 D 3 E infinitely many Solution: B ? 2 ? ? 7 3 ? 4 ? N ? 5 ? ?

9. Example
“Show that the sum of any three consecutive integers is a multiple of 3.”
To show it’s true for any three consecutive numbers, we need to generically represent any possible three consecutive numbers!
The first number 𝑛 could be any integer, hence keeping our proof general (i.e. not specific to a particular case). But by using 𝑛+1 and 𝑛+2, we ensure that the numbers are consecutive.
Let the numbers be:
𝑛, 𝑛+1, 𝑛+2
Then sum of numbers is:
𝑛+𝑛+1+𝑛+2 =3𝑛+3
The questions asks for the sum of these numbers.
Factorise out (rather than divide by) the 3 to show our expression is explicitly 3 times an integer, and therefore a multiple of 3.
=3(𝑛+1)
which is divisible by 3.
Explicitly point this out to complete your proof.

10. “Show that the sum of any four consecutive integers is even.”
Test Your Understanding So Far
“Show that the sum of any four consecutive integers is even.”
Hint: Anything that is even is a multiple of 2. ?
Let the numbers be:
𝒏, 𝒏+𝟏, 𝒏+𝟐, 𝒏+𝟑
The sum of numbers is:
𝒏+𝒏+𝟏+𝒏+𝟐+𝒏+𝟑=𝟒𝒏+𝟔 =𝟐(𝟐𝒏+𝟑)
which is even.

11. ? ? ? ? ? ? ? Representing other numbers !
All odd numbers are 1 more than a multiple of 2.
How could I algebraically represent: !
Consecutive integers
𝑛, 𝑛+1
An odd number
2𝑛+1
An even number 2𝑛
Two consecutive odd numbers
2𝑛+1, 𝑛+3
(but 2𝑛−1 and 2𝑛+1 might make your subsequent algebra easier)
Two odd numbers
2𝑛+1, 2𝑚+1
One less than a multiple of 5.
5𝑛−1
A number that when divided by 4, gives you a remainder of 2.
4𝑛+2 ? ? ? ? ? ? ?
It would be wrong to use 2𝑛+1 and 2𝑛+3 because this restricts our odd numbers only to consecutive ones. By using a different variable, we allow the second odd number to be different from the first.

12. ? Further Examples Let the numbers be 2𝑛, 2𝑛+2, 2𝑛+4
“Prove that the sum of three consecutive even numbers is a multiple of 6.” ?
Let the numbers be 2𝑛, 2𝑛+2, 2𝑛+4
Then: 2𝑛+2𝑛+2+2𝑛+4=6𝑛+6 =6(𝑛+1)
which is a multiple of 6.

13. Harder Example Let the numbers be 2𝑛+1, 2𝑛+3
“Prove that the product of two consecutive odd numbers is always one less than a multiple of 4.”
Let the numbers be 2𝑛+1, 2𝑛+3
Then: 2𝑛+1 2𝑛+3 =4 𝑛 2 +6𝑛+2𝑛+3 =4 𝑛 2 +8𝑛+3 = −1
which is 1 less than a multiple of 4.
𝑛 2 +2𝑛+1
At this stage, you might be tempted to write =4𝑛−1 to represent “1 less than a multiple of 4”. But you’ve already used 𝒏 earlier to represent the original numbers.
Instead we think “4 times something – 1”, and therefore write 4 … −1 before working out what goes in the bracket.

14. “Prove that the product of two odd numbers is odd.”
Harder Example
“Prove that the product of two odd numbers is odd.”
Recall that 2𝑛+1 and 2𝑛+3 would be wrong, because this would be restricting our proof to consecutive odd numbers only, not any two odd numbers.
Let the numbers be: 2𝑛+1, 2𝑚+1
Then:
2𝑛+1 2𝑚+1 =4𝑚𝑛+2𝑚+2𝑛+1 =
which is odd.
2𝑚𝑛+𝑚+𝑛
An odd number can be written in the form 2 … +1. Remember we can’t write 2𝑛+1 or 2𝑚+1 because we’ve already used 𝑚 and 𝑛.

15. Test Your Understanding
“Prove that the sum of three consecutive odd numbers is 3 more than a multiple of 6.” ?
Let the numbers be:
𝟐𝒏+𝟏, 𝟐𝒏+𝟑, 𝟐𝒏+𝟓
Then:
𝟐𝒏+𝟏+𝟐𝒏+𝟑+𝟐𝒏+𝟓 =𝟔𝒏+𝟗 =𝟔 𝒏+𝟏 +𝟑
which is 3 more than a multiple of 6.

16. “Prove that the sum of the squares of two consecutive numbers is odd.”
Proofs involving Squares
“Prove that the sum of the squares of two consecutive numbers is odd.” ?
Let the numbers be:
𝒏, 𝒏+𝟏
Then:
𝒏 𝟐 + 𝒏+𝟏 𝟐 = 𝒏 𝟐 + 𝒏 𝟐 +𝟐𝒏+𝟏 =𝟐 𝒏 𝟐 +𝟐𝒏+𝟏 =𝟐 𝒏 𝟐 +𝟐𝒏 +𝟏
which is odd.

17. (Solutions on next slides)
Exercise
(Solutions on next slides)
Prove that the sum of 5 consecutive integers is a multiple of 5.
Show algebraically that the sum of any 3 consecutive even numbers is always a multiple of 6.
Prove that the sum of any two even numbers is even.
Prove that the difference between the squares of two consecutive odd numbers is a multiple of 8.
Prove that the sum of the squares of any three consecutive odd numbers is always 11 more than a multiple of 12.
Prove algebraically that the difference between any two different odd numbers is an even number.
Prove algebraically that the sum of the squares of any two odd numbers leaves a remainder of 2 when divided by 4.
Prove algebraically that the difference between the squares of any two consecutive integers is equal to the sum of these two integers.
Prove that the difference between two consecutive cubes is one more than a multiple of 6.
[UKMT] Prove that there is exactly one sequence of five consecutive positive integers in which the sum of the squares of the first three integers is equal to the sum of the squares of the other two integers.
Prove that the product of four consecutive numbers is one less than a square number. 1 7 2 8 3 N1 4 N2 5 6 N3

18. Solutions
Prove that the sum of 5 consecutive integers is a multiple of 5.
𝒏+𝒏+𝟏+𝒏+𝟐+…+𝒏+𝟒 =𝟓𝒏+𝟏𝟎=𝟓(𝒏+𝟐)
which is a multiple of 5.
Show algebraically that the sum of any 3 consecutive even numbers is always a multiple of 6.
𝟐𝒏+𝟐𝒏+𝟐+𝟐𝒏+𝟒 =𝟔𝒏+𝟔=𝟔(𝒏+𝟏)
which is a multiple of 6.
Prove that the sum of any two even numbers is even.
𝟐𝒎+𝟐𝒏=𝟐(𝒎+𝒏)
which is even.
Prove that the difference between the squares of two consecutive odd numbers is a multiple of 8.
𝟐𝒏+𝟑 𝟐 − 𝟐𝒏+𝟏 𝟐 = 𝟒 𝒏 𝟐 +𝟏𝟐𝒏+𝟗 − 𝟒 𝒏 𝟐 +𝟒𝒏+𝟏 =𝟖𝒏+𝟖=𝟖(𝒏+𝟏)
which is a multiple of 8.
Prove that the sum of the squares of any three consecutive odd numbers is always 11 more than a multiple of 12.
𝟐𝒏+𝟏 𝟐 + 𝟐𝒏+𝟑 𝟐 + 𝟐𝒏+𝟓 𝟐 =𝟏𝟐 𝒏 𝟐 +𝟐𝟒𝒏+𝟑𝟓 =𝟏𝟐 𝒏 𝟐 +𝟐𝒏+𝟐 +𝟏𝟏
which is 11 more than a multiple of 12. 1 4 ? ? 2 ? 5 ? 3 ?

19. Solutions 6
Prove algebraically that the difference between any two different odd numbers is an even number.
𝟐𝒎+𝟏 − 𝟐𝒏+𝟏 =𝟐𝒎−𝟐𝒏 =𝟐(𝒎−𝒏)
which is even.
Prove algebraically that the sum of the squares of any two odd numbers leaves a remainder of 2 when divided by 4.
𝟐𝒎+𝟏 𝟐 + 𝟐𝒏+𝟏 𝟐 =𝟒 𝒎 𝟐 +𝟒𝒎+𝟏+𝟒 𝒏 𝟐 +𝟒𝒏+𝟏 =𝟒 𝒎 𝟐 +𝟒𝒎+𝟒 𝒏 𝟐 +𝟒𝒏+𝟐 =𝟒 𝒎 𝟐 +𝒎+ 𝒏 𝟐 +𝒏 +𝟐
which is 2 more than a multiple of 4.
Prove algebraically that the difference between the squares of any two consecutive integers is equal to the sum of these two integers.
Let numbers be 𝒏, 𝒏+𝟏
Difference of squares:
𝒏+𝟏 𝟐 − 𝒏 𝟐 = 𝒏 𝟐 +𝟐𝒏+𝟏− 𝒏 𝟐 =𝟐𝒏+𝟏
Sum:
𝒏+ 𝒏+𝟏 =𝟐𝒏+𝟏
These are equal. 7 ? ?

20. Solutions
Prove that the difference between two consecutive cubes is one more than a multiple of 6.
𝒏+𝟏 𝟑 − 𝒏 𝟑 = 𝒏 𝟑 +𝟑 𝒏 𝟐 +𝟑𝒏+𝟏− 𝒏 𝟑 =𝟑 𝒏 𝟐 +𝟑𝒏+𝟏 =𝟑𝒏 𝒏+𝟏 +𝟏
The product of two consecutive integers is even, thus 𝟑𝒏(𝒏+𝟏) is divisible by 6. N1
[IMO] Prove that there is exactly one sequence of five consecutive positive integers in which the sum of the squares of the first three integers is equal to the sum of the squares of the other two integers.
𝒙−𝟏 𝟐 + 𝒙 𝟐 + 𝒙+𝟏 𝟐 = 𝒙+𝟐 𝟐 + 𝒙+𝟑 𝟐 𝟑 𝒙 𝟐 +𝟐= 𝟐𝒙 𝟐 +𝟏𝟎𝒙+𝟏𝟑 𝒙 𝟐 −𝟏𝟎𝒙−𝟏𝟏=𝟎 𝒙+𝟏 𝒙−𝟏𝟏 =𝟎
𝒙=−𝟏 𝒐𝒓 𝒙=𝟏𝟏
So 𝟏𝟎 𝟐 + 𝟏𝟏 𝟐 + 𝟏𝟐 𝟐 = 𝟏𝟑 𝟐 + 𝟏𝟒 𝟐 is only solution.
Prove that the product of four consecutive numbers is one less than a square number.
One more than the product of four consecutive integers should be a square:
𝒂 𝒂+𝟏 𝒂+𝟐 𝒂+𝟑 +𝟏 = 𝒂 𝟐 +𝒂 𝒂 𝟐 +𝟓𝒂+𝟔 +𝟏 = 𝒂 𝟒 +𝟔 𝒂 𝟑 +𝟏𝟏 𝒂 𝟐 +𝟔𝒂+𝟏 = 𝒂 𝟐 +𝟑𝒂+𝟏 𝟐 N2 ? ? N3 ?

21. An Oxford entrance test question (yes, really!)
Source: MAT A   B  C  D
The statement “If 𝐴, then 𝐵” is only falsified when 𝐴 is true but 𝐵 is false. If the condition of the statement is not true, then the statement is not applicable, and we can neither say it is true or false.