Proof & Number Theory

4 methods · 12 questions

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A claim of the form 'for all $n$ , P(n)' or 'if $n$ is prime then Q(n)' is refuted by a single case that satisfies the hypothesis yet breaks the conclusion. When the candidates are handed to you (I, II, III) or are small, do not prove anything: substitute and check. Watch the hypothesis first, a case that fails the hypothesis can never be a counterexample.

Trigger: 'for all', 'always', 'is a counterexample', or a list of explicit candidate values I/II/III.

Instances:
(i) plug each listed value into both sides of an identity and keep the one that disagrees;
(ii) for 'if prime then ...', test the small primes $2, 3, 5$ and discard any candidate whose number is not prime;
(iii) for a claim about a family, find the one value where the divisibility or primality pattern breaks.

Linked questions (3)

TMUA 2016 Paper 2 · Q5

Consider the statement:

(*)

A whole number

n

is prime if it is 1 less or 5 less than a multiple of 6.

How many counterexamples to

(*)

are there in the range

0 < n < 50

TMUA 2022 Paper 2 · Q3

Consider the following statement about the positive integer

n

[[B]] i f [[/ B]] n is prime, [[B]] t h e n [[/ B]] n^{2} + 2 is [[B]] n o t [[/ B]] prime

Which of the following is a counterexample to this statement?

I: $n = 2$
II: $n = 3$
III: $n = 4$

Anone of them
BI only
CII only
DIII only
EI and II only
FI and III only
GII and III only
HI, II and III

TMUA 2023 Paper 2 · Q3

Consider the claim: for all positive real numbers

x

and

y

x^{y} = x^{y} .

Which of the following is/are a counterexample to the claim? I:

x = 1, y = 16

. II:

x = 2, y = 8

. III:

x = 3, y = 4

Anone of them
BI only
CII only
DIII only
EI and II only
FI and III only
GII and III only
HI, II and III