(For the following problem I don't just want a flat out answer, but steps and Ideas on how to solve it. The problem was given by my Universities newspaper and for solving it you get free Loot and stuff) ---------------------------------------------------------------------------------------------------------------------------------------- The Problem: Show that every set of n+1 positive integers, chosen from a set of 2n consecutive integers, contains at least one pair of relatively prime numbers. relatively prime means: "Two integers are relatively prime if they share no common positive factors (divisors) except 1" Example: 8 and 15 are relatively prime. 8 = 2 * 2 * 2 15 = 3 * 5 Example: 9 and 12 are NOT relatively prime 9 = 3 * 3 12 = 2 * 2 * 3 ---------------------------------------------------------------------------------------------------------------------------------------- My thinking thus far: None really I've just started
very true, but I think the problem is more difficult than this. "Show that every set of n+1 positive integers, chosen from a set of 2n consecutive integers" wouldn't this mean the set would be... n= 0 1 2 3 4 Set=0,2,6,8 then you would do the n+1 one on that set? Or something to that extent?
Petek's hint is right. If you wanted to choose a subset without picking any pair that's relatively prime, what does that tell you about the option of picking consecutive numbers from the set?
The statement means: I give you a set S = {a, a+1, a+2, ..., a+2n-1} and you pick a subset with n+1 elements. The point of the problem is that no matter what you do, your subset always contains a pair which are relatively prime.