00:01
In this problem, we're given the set of positive integers from 1 through 100, we're calling that set a, and we choose a subset b from that set that contains exactly 51 elements.
00:12
Our job is to prove that we can find two elements x and y in the subset b, such that x and y are co -prime, that is their greatest common divisor, is 1.
00:24
So the key result we want to take advantage of here is the following fact.
00:29
That for any positive integer in, we know that the greatest common divisor of n and n plus 1 is 1.
00:54
So we are arguing that consecutive integers are always coprined.
00:59
And the reason for this is any gcd of these two have to divide their difference also, and their difference is n plus 1 minus n or 1.
01:07
Therefore, their greatest common divisor has to be 1.
01:10
So if we pick 51 elements out of the given set, which contains 100 elements consecutive, then we're going to get at least one consecutive pair.
01:21
So that's our main argument here.
01:25
So let's just copy a over again.
01:28
Our set a contains the positive integers from 1 up to 100.
01:39
And let's choose a subset b out of a that contains 51 elements...