Est. 1998 — 27 years of woodworking knowledge
Friday Puzzler -- Relative Primes
Alex Y
Two numbers are said to be relatively prime if they have no common divisors. E.g., 10 and 21 are relatively prime, since while both are composite numbers, they have no divisors in common.
For a set of ten consecutive positive integers, what are the possible values for the number of integers in the set that are relatively prime to all other numbers in the set?
Re: Friday Puzzler -- Relative Primes
Larry Barrett
In any set of 10 consecutive integers, half will be even so the number of relative primes can not be greater than 5. Of the remaining odd integers, there will be a multiple of 5, which will not be prime relative to the multiple of 10 in the set, so the multiple of 5 can also be eliminated. So of the remaining 4 odd integers in the original set of 10, how many of these will be prime relative to the original set? So far 4, 3, and 2 are possible; I have not found examples of 1. A proof seems possible, though I have not tried to create one (too nice outside today).
On the right track
Re: Friday Puzzler -- Relative Primes
Alex Y
Larry, good job on this one. The first example of ten consecutive positive integers, only one of which is relatively prime to all the others, is 210, 211, 212, ..., 219.
211 is relatively prime to all the others.