TMUA Official 2023 Paper 2 · Question 11 · ExamDay

Official papers

Official 2023 Paper 2 · Q11

Paper 2Logic & proofNumber & counting

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 $(*)$ ?

I. If $k$ is a power of 2, then $2^k+1$ is prime.
II. $2^k+1$ 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

ABCDEFGH

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