Proof - Year 1 Core PhysicsAndMathsTutor.

com
PMT

1. A student is investigating the following statement about natural numbers.

“ n 3 – n is a multiple of 4 ”

(a) Prove, using algebra, that the statement is true for all odd numbers.
(4)
(b) Use a counterexample to show that the statement is not always true.
(1)
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Proof - Year 1 Core PhysicsAndMathsTutor.com
PMT

2. (i) A student states

“if x 2 is greater than 9 then x must be greater than 3”

Determine whether or not this statement is true, giving a reason for your answer.
(1)
(ii) Prove that for all positive integers n,

n3 + 3n 2 + 2n

is divisible by 6
(3)
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*P69201A04648* 
Proof - Year 1 Core PMT
PhysicsAndMathsTutor.com

3.
17. In this question p and q are positive integers with q > p
Statement 1: is never a multiple of 5

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(a) Show, by means of a counter example, that Statement 1 is not true.
(1)
Statement 2: When p and q are consecutive even integers is a multiple of 8
(b) Prove, using algebra, that Statement 2 is true.
(4)
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*P72839A04244* 
Proof - Year 1 Core PhysicsAndMathsTutor.com
PMT

4. (i) Use proof by exhaustion to show that for n ∈ , n  4

3 n
(n + 1) > 3
(2)
(ii) Given that m 3 + 5 is odd, use proof by contradiction to show, using algebra, that m
is even.
(4)
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*P68731A04852* 
Proof - Year 1 Core PhysicsAndMathsTutor.com
PMT

5. Prove, using algebra, that

(n + 1)3 – n3

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is odd for all n ∈ 
(4)
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