A student attempts to answer the following question.
What is the largest number of consecutive odd integers that are all prime?
The student's attempt is as follows:
I There are two consecutive odd integers that are prime (for example: 17, 19).
II Any three consecutive odd integers can be written in the form n−2, n, n+2 for some n.
III If n is one more than a multiple of 3, then n+2 is a multiple of 3.
IV If n is two more than a multiple of 3, then n−2 is a multiple of 3.
V The only other possibility is that n is a multiple of 3.
VI In each case, one of the integers is a multiple of 3, so not prime.
VII Therefore the largest number of consecutive odd integers that are all prime is two.
Which of the following best describes this attempt?
- AIt is completely correct.
- BIt is incorrect, and the first error is on line I.
- CIt is incorrect, and the first error is on line II.
- DIt is incorrect, and the first error is on line III.
- EIt is incorrect, and the first error is on line IV.
- FIt is incorrect, and the first error is on line V.
- GIt is incorrect, and the first error is on line VI.
- HIt is incorrect, and the first error is on line VII.
Show the answer and worked solution
answer · G
- AIt is completely correct.
- BIt is incorrect, and the first error is on line I.
- CIt is incorrect, and the first error is on line II.
- DIt is incorrect, and the first error is on line III.
- EIt is incorrect, and the first error is on line IV.
- FIt is incorrect, and the first error is on line V.
- GIt is incorrect, and the first error is on line VI.
- HIt is incorrect, and the first error is on line VII.
Lines I to V are all correct: three consecutive odd numbers do take the form n−2, n, n+2, and the case analysis on n modulo 3 is exhaustive and right. Line VI is where it fails — being a multiple of 3 does not stop a number being prime, because 3 itself is prime. The triple 3, 5, 7 is three consecutive odd integers that are all prime, which also makes the conclusion in line VII wrong.