Worked examples
The question
Question: prove that the sum of two even numbers is even.
1 Write it with letters
x is even, so x = 2a, where a is a whole number.
x = 2a
That is what even means.
even = 2 × a whole number
y is even, so y = 2b, where b is a whole number.
A new letter, b: y can be a different number from x.
y = 2b
Careful!
The same letter would make x and y the same number.
Two numbers need two letters.
Careful: not x = 2a, y = 2a ✗
2 Put them in
Now put them in: x + y becomes (2a) + (2b).
x + y = (2a) + (2b)
3 Work it out
Open the brackets: 2a + 2b.
= 2a + 2b
4 Show the form
Every term has a 2 in it.
We take out 2: 2(a + b).
= 2(a + b)
a + b is a whole number, because whole numbers added and multiplied stay whole.
a + b is a whole number
2 times a whole number: so x + y is even.
So x + y is even. ∎
The square at the end means: the proof is done.
5 Check
An example: x = 2 and y = 4.
It gives 6.
x = 2, y = 4: 2 + 4 = 6
6 is 2 times 3: even, as the proof says.
Right!
6 = 2 × 3
Let's look back.
First: write it with letters.
Then: put them in.
Then: work it out.
Then: show the form.
Last: check.
Answer a + b is a whole number
▶ Watch it solved
The question
Question: prove that the product of two odd numbers is odd.
1 Write it with letters
x is odd, so x = 2a + 1, where a is a whole number.
x = 2a + 1
That is what odd means.
odd = 2 × a whole number + 1
y is odd, so y = 2b + 1, where b is a whole number.
A new letter, b: y can be a different number from x.
y = 2b + 1
Careful!
The same letter would make x and y the same number.
Two numbers need two letters.
Careful: not x = 2a + 1, y = 2a + 1 ✗
2 Put them in
Now put them in: xy becomes (2a + 1)(2b + 1).
xy = (2a + 1)(2b + 1)
3 Work it out
Every part times every part: 4ab + 2a + 2b + 1.
= 4ab + 2a + 2b + 1
4 Show the form
We take out 2 from all but the 1: 2(2ab + a + b) + 1.
= 2(2ab + a + b) + 1
2ab + a + b is a whole number, because whole numbers added and multiplied stay whole.
2ab + a + b is a whole number
2 times a whole number, plus 1: that is the form of an odd number.
So xy is odd.
So xy is odd. ∎
The square at the end means: the proof is done.
5 Check
An example: x = 3 and y = 5.
It gives 15.
x = 3, y = 5: 3 × 5 = 15
15 is 2 times 7, plus 1: odd, as the proof says.
Right!
15 = 2 × 7 + 1
Let's look back.
First: write it with letters.
Then: put them in.
Then: work it out.
Then: show the form.
Last: check.
Answer 2ab + a + b is a whole number
▶ Watch it solved
The question
Question: prove that if n is odd, then n² + n + 1 is odd.
1 Write it with letters
n is odd, so n = 2a + 1, where a is a whole number.
n = 2a + 1
That is what odd means.
odd = 2 × a whole number + 1
Careful!
An example shows one number.
The letters show every number at once.
Careful: one example is not a proof
2 Put them in
Now put them in: n² + n + 1 becomes (2a + 1)² + (2a + 1) + 1.
n2 + n + 1 = (2a + 1)2 + (2a + 1) + 1
3 Work it out
(2a + 1)² means the bracket times itself.
= (2a + 1)(2a + 1) + (2a + 1) + 1
Every part times every part: 4a² + 2a + 2a + 1 + 2a + 1 + 1.
= 4a2 + 2a + 2a + 1 + 2a + 1 + 1
Like terms together: 4a² + 6a + 3.
= 4a2 + 6a + 3
4 Show the form
We take out 2 from all but the 1: 2(2a² + 3a + 1) + 1.
= 2(2a2 + 3a + 1) + 1
2a² + 3a + 1 is a whole number, because whole numbers added and multiplied stay whole.
2a² + 3a + 1 is a whole number
2 times a whole number, plus 1: that is the form of an odd number.
So n² + n + 1 is odd.
So n² + n + 1 is odd. ∎
The square at the end means: the proof is done.
5 Check
An example: n = 3.
It gives 13.
n = 3: 3² + 3 + 1 = 13
13 is 2 times 6, plus 1: odd, as the proof says.
Right!
13 = 2 × 6 + 1
Let's look back.
First: write it with letters.
Then: put them in.
Then: work it out.
Then: show the form.
Last: check.
Answer 2a² + 3a + 1 is a whole number
▶ Watch it solved
