Find counterexamples to the following statements.
- (i) If n is a prime number, then 2n − 1 is a prime number.
- (ii) If n is an even number, then 2n + 1 is a prime number.
Answer: (i) n = 11 (prime): 211 − 1 = 2047 = 23 × 89, not prime. (ii) n = 6 (even): 26 + 1 = 65 = 5 × 13, not prime.
Step-by-step solution
Idea: A counterexample must make the ‘if’ part true and the ‘then’ part false: in (i) a prime n with 2n − 1 composite; in (ii) an even n with 2n + 1 composite.
(i) If n is a prime number, then 2n − 1 is a prime number.
- Primes n = 2, 3, 5, 7 give 3, 7, 31, 127 — all prime. The next prime, n = 11, gives 211 − 1 = 2048 − 1 = 2047.½ mark
- 2047 = 23 × 89 is not prime. So n = 11 is a counterexample.½ mark
n = 11: 211 − 1 = 2047 = 23 × 89, not prime.
(ii) If n is an even number, then 2n + 1 is a prime number.
- Even n = 2, 4 give 5 and 17 — prime. n = 6 gives 26 + 1 = 64 + 1 = 65.½ mark
- 65 = 5 × 13 is not prime. So n = 6 is a counterexample.½ mark
n = 6: 26 + 1 = 65 = 5 × 13, not prime.
(i) n = 11: 211 − 1 = 2047 = 23 × 89. (ii) n = 6: 26 + 1 = 65 = 5 × 13.
Check: 23 × 89 = 2047 ✓ and 5 × 13 = 65 ✓.
Answer to write in the exam
(i)
n = 11 is prime.
211 − 1 = 2048 − 1 = 2047 = 23 × 89
∴ n = 11 is a counterexample.
(ii)
n = 6 is even.
26 + 1 = 64 + 1 = 65 = 5 × 13
∴ n = 6 is a counterexample.
Common mistakes that cost marks
- Testing only n = 2, 3, 5, 7 in (i) and concluding the statement is true. The first failure is at n = 11.
- Using a non-prime n in (i), e.g. n = 4 (24 − 1 = 15). The ‘if’ part needs n to be prime, so this is not a counterexample.
- Using an odd n in (ii); the ‘if’ part needs n to be even.
How this can come in the exam
MCQ (1 mark)
Which of these numbers is NOT prime?
- 27 − 1
- 213 − 1
- 211 − 1
- 25 − 1
Show answer
(C) 211 − 1
211 − 1 = 2047 = 23 × 89. The others are 127, 8191 and 31, all prime.
Short answer (2 marks)
Show that 29 − 1 is not prime. Is n = 9 a counterexample to ‘If n is a prime number, then 2n − 1 is a prime number’?
Show answer
29 − 1 = 511 = 7 × 73, so it is not prime (1 mark). It is not a counterexample: 9 is not prime, so the ‘if’ part is false (1 mark).Try one yourself
Show that 210 + 1 is not prime.
Show answer
210 + 1 = 1025, which ends in 5, so it is divisible by 5: 1025 = 5 × 205.
More questions like this
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Suppose we have identified a category of quadrilaterals called Q, and we have to construct a quadrilateral of this type. For this, we are to use two thin sticks, put them together as diagonals so that the quadrilateral obtained by joining their endpoints is of type Q (see the figure). - If a proposition is true, then is its converse always true?
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Statement 2: If two angles of a triangle are equal, then the sides opposite the equal angles have equal lengths.
We have proved the first statement in an earlier grade. Is the second statement true? Can you prove it?
(Hint: Draw the altitude from the vertex containing the third angle.)