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Problem 485

AMC 10/12, early questions

Number theory Difficulty 3.0 Prove it BMO Round 1 · United Kingdom · 2020

Show that there are at least three prime numbers $p$ less than $200$ for which $p + 2$ , $p + 6$ , $p + 8$ and $p + 12$ are all prime. Show also that there is only one prime number $q$ for which $q + 2$ , $q + 6$ , $q + 8$ , $q + 12$ and $q + 14$ are all prime.

This one wants a proof. Work it on paper, then check yourself against the publisher's own solution, linked below. Be honest about it: the record is only any use to you if it is.

We don't reproduce this publisher's solutions. Their own solution is here — work the problem first.

Source: UK Mathematics Trust, licensed © UK Mathematics Trust; question papers published free at bmos.ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project. Solutions are the publisher's, linked not copied.