Full 360
Probability
Level
1
Is it true that in any eight composite positive integers not exceeding 360, that at least two are not relatively prime?
False
True
There is insufficient information
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1 solution
Moderator note:
Simple standard approach with the pigeonhole principle.
Proof(by contradiction): Suppose there exist eight positive composite integers not exceeding 360 that are pairwise relatively prime. We observe that any positive composite integer not exceeding 360 must have a prime divisor not exceeding 17. That is, each of our eight positive composite integers must be divisible by one of the seven primes 2, 3, 5, 7, 11, 13, or 17. As there are eight composite integers and seven primes, by the Pigeonhole Principle, at least two of our eight integers must be divisible by the same prime and hence have a greatest common divisor greater than 1. This is a contradiction.