TMUA 2023 Paper 2 Q11 — Worked Solution

P2·Logic & Quantifiers, Number Theory & Divisibility·6.5Official · 2023 · Q11

logical-equivalenceconverse-contrapositivefermat-primes

In this question,

k

is a positive integer.Consider the following theorem:If

2^{k} + 1

is a prime, then

k

is a power of

2

(*)

Which of the following statements, taken individually, is/are equivalent to

(*)

k

is a power of

2

, then

2^{k} + 1

is prime.

2^{k} + 1

is not prime only if

k

is not a power of

2

III

A sufficient condition for

k

to be a power of

2

is that

2^{k} + 1

is prime.

	Statement I is equivalent to $(*)$	Statement II is equivalent to $(*)$	Statement III is equivalent to $(*)$
A	Yes	Yes	Yes
B	Yes	Yes	No
C	Yes	No	Yes
D	Yes	No	No
E	No	Yes	Yes
F	No	Yes	No
G	No	No	Yes
H	No	No	No

Discussion

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	Statement I is equivalent to $(*)$	Statement II is equivalent to $(*)$	Statement III is equivalent to $(*)$
A	Yes	Yes	Yes
B	Yes	Yes	No
C	Yes	No	Yes
D	Yes	No	No
E	No	Yes	Yes
F	No	Yes	No
G	No	No	Yes
H	No	No	No

	Statement I is equivalent to $(*)$	Statement II is equivalent to $(*)$	Statement III is equivalent to $(*)$
A	Yes	Yes	Yes
B	Yes	Yes	No
C	Yes	No	Yes
D	Yes	No	No
E	No	Yes	Yes
F	No	Yes	No
G	No	No	Yes
H	No	No	No

	Statement I is equivalent to $(*)$	Statement II is equivalent to $(*)$	Statement III is equivalent to $(*)$
A	Yes	Yes	Yes
B	Yes	Yes	No
C	Yes	No	Yes
D	Yes	No	No
E	No	Yes	Yes
F	No	Yes	No
G	No	No	Yes
H	No	No	No