Download A. Counter examples 1. Brian says all prime numbers are odd. Prove

Proof
To prove a statement we need to give an accurate mathematical argument.
This could involve many mathematical techniques, usually algebra.
To prove a statement is false, finding one counter example is sufficient
A. Counter examples
1. Brian says all prime numbers are odd. Prove he is incorrect.
2 is a prime number and 2 is even
2 is a counter example
2. Helen says all odd numbers are prime. Prove Helen is incorrect.
9 is an odd number and 9 is not prime because it has more than 2 factors.
The factors of 9 are 1, 3, 9
9 is a counter example
3. Colin says if x and y are prime numbers will x2 + y2 will be an even number. Is he correct?
If x = 3 and y = 5
32 + 52 = 9 + 25 = 34 34 is even
2
2
If x = 11 and y = 7
11 + 7 = 121 + 49 = 170 170 is even
If x = 2 and y = 5
22 + 52 = 4 + 25 = 29 29 is odd
Try some examples to see
if you can find a counter
example
It just takes one counter example
to prove he is wrong
He is incorrect
4. Sandra says the cube of a number is always larger than its square. Is she correct?
If x = 3
If x = 1
If x = -2
32 = 9
2
1 =1
-22 = 4
33 = 27 true
This counter example shows the
cube is the same as the square
3
1 = 1 false
-23 = -8 false
This counter example shows the cube is
the smaller than the square
She is incorrect
B. Algebraic proof
Quite often the easiest way to prove a statement is to use algebra
*To choose 2 consecutive numbers using algebra we could use n and n + 1, however we
would not know which one was odd and which was even.
*To choose an even number we could use 2n, as doubling any number will make it even.
*Any two even numbers could be 2n and 2m.
*As 2n will be even using 2n + 1 or 2n – 1 will give us an odd number
*2n is a multiple of 2, 3n is a multiple of 3, 4n is a multiple of 4, etc.
1. Prove the sum of three consecutive numbers will be a multiple of 3
Let the numbers be n, n + 1, n + 2
Start by showing the numbers you are going to use
n + n + 1 + n + 2 = 3n + 3
= 3(n + 1)
As we could divide by 3 the sum has to be a multiple of 3
2. Prove the sum of two consecutive numbers is odd
Let the numbers be n, n + 1
Start by showing the numbers you are going to use
n + n + 1 = 2n + 1
2n + 1 can’t be divided exactly by 2 so the sum must be odd
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Proof
3. Prove the product of two consecutive numbers is even
Let the numbers be n, n + 1
Show the numbers you are going to use
Product = n(n + 1) = n2 + n
We cannot show that we can divide by 2
Choosing a different start may help!
The two numbers must be of the form 2n, 2n+1 or 2n-1, 2n depending on whether the
smaller number is even or odd
Choose new starting values
Let the numbers be 2n, 2n + 1
We have now shown we can divide by 2,
so the product must be even
Product = 2n(2n + 1) = 4n2 + 2n
= 2(2n2 + n)
It is easy to show that the same approach will work with 2n-1, 2n. Check this.
4. Show that the sum of three consecutive even numbers will be a multiple of 6
Let the numbers be 2n, 2n + 2, 2n + 4
Show the numbers you are going to use
Sum = 2n + 2n + 2 + 2n + 4
= 6n + 6
As we could divide by 6 the sum has to be a
= 6(n + 1)
multiple of 6
5. Show that the difference of two consecutive square numbers is odd
Let the numbers be n, n + 1
Start by showing the numbers you are going to
2
2
use, then square them
n and (n + 1) = (n + 1)(n + 1)
2
=n +n+n+1
= n2 + 2n + 1
2
Difference = n + 2n + 1 - n2
= 2n + 1
2n has to be even, so 2n + 1 must be odd
6. Prove that (n + 3)2 ≡ (n + 2)(n + 4) + 1
Show the
LHS = RHS
(n + 3)2
≡ (n + 3)(n + 3)
≡ n2 + 3n + 3n + 9
≡ n2 + 6n + 8 + 1
≡ (n+2)(n+4) + 1
Both sides are the same
7. Prove that (2n + 1)2 – (2n - 1)2 will be a multiple of 8
Square each bracket
= (2n + 1)(2n + 1) – (2n - 1)(2n – 1)
2
2
= (4n + 2n + 2n + 1) – (4n - 2n – 2n + 1)
= (4n2 + 4n + 1) – (4n2 - 4n + 1)
Be careful with the signs when doing the
subtraction
= 4n2 + 4n + 1 – 4n2 + 4n – 1
= 8n and 8n is a multiple of 8 as it can be divided by 8
8. Prove that a = 3 and b = -2 when 3(ax + 2) – 2(x - b) ≡ 7x + 4
3ax + 6 – 2x + b ≡ 7x + 4
x(3a – 2) + 6 + b ≡ 7x + 4
3a – 2 = 7
6+b=4
3a = 9
b=4–6
𝟗
a=𝟑
a=3
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Multiply out the
brackets then equate
the coefficients
b = -2
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