Worksheet: Proof CM
1 (a) Prove that x2 + 4x + 12 > 0 for all integers x.
(b) It is claimed that “n2 + 3n + 1 > 0 for all integers n”
Disprove this statement using a suitable counter-example.
2 Emily wants to prove that the sum of the squares of any two odd numbers is even.
Her proof is shown below.
Let the odd numbers be 2n – 1 and 2n + 1 for integers n.
Then
(2n – 1)2 + (2n + 1)2 = 4n2 – 4n + 1 + 4n2 + 4n + 1
= 8n2 + 2
= 2(4n2 + 1)
which is a multiple of 2 and so even.
Hence, the sum of the squares of any two odd numbers is even.
Her proof is not correct.
(a) Explain why.
(b) Provide a correct proof of the statement.
3 Let p and q be prime numbers, p, q ≠ k, where k is an integer.
Given that p + q is always even,
(a) state the value of k.
(b) Prove that p + q is always even for all primes except k.
4 The claim is that if n is an integer, then q = n2 – 2 is not divisible by 4.
(a) Use exhaustion to prove that the statement is true for 4 ≤ n ≤ 7.
[The rest of the question is A level only. It is broken down since it is a bit challenging.]
Now we will prove the claim in full.
(b) Consider the case when n is odd. Is n2 odd or even? What about n2 – 2?
Hence, explain why it cannot be a multiple of 4 in this case.
(c) Use a similar process for the case when n is even to show the claim is not true in this case
either.
5 (a) Prove that if n is even, then (3n)2 is even.
(b) Prove that if (3n)2 is even, then n is even.
[Hint for (b): consider factors]
6 [A level only]
For all real x, we have that
(5x – 3)2 + 1 ≥ (3x – 1)2
Prove the statement using a proof by contradiction.
7 [A level only]
(a) Prove that the square root of 2 is irrational.
(b) Prove that the square root of 3 is irrational.
(c) Prove that the cube root of 5 is irrational.
8 [A level only]
Prove that there are an infinite number of primes.
9 [A level only]
25
(a) Prove that, for all real and positive x, x + ≥ 10
x
(b) Give a counter-example to show that the statement is not necessarily true if x is real but
negative.
(c) Is the statement ever true if x is negative?
10 Consider the statement
“If 3n2 + 2n is even, then n is even”
In this question, you will prove this in three ways.
Method 1:
(a) Since 3n2 + 2n is even, it can be written in the form 3n2 + 2n = 2k for some integer k.
By considering factors, complete the proof.
Method 2:
(b) Write 3n2 + 2n = n2 + (2n2 + 2n) = 2k for some integer k.
Think about what you can deduce about n2 and complete the proof.
Method 3 [A level only]:
(c) Prove the statement by contradiction.
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