I have been reading this problem,
Prove that of any 52 integers, two can always be found such that the difference of their squares is divisible by 100.
Now, in my understanding, it says, any 52 integers.
So I could just choose 0 and 100, from a random set of integers right?
100^2 - 0^2 = 10000 which is devisible by 100.
What I am not getting here? I read the answers but I still cant figure out what exactly the problem is asking.
Basically I am trying to understand the first answer:
Look at your 52 integers mod 100. Look at the pairs of additive inverses (0,0), (1,99), (2,98), etc. There are 51 such pairs. Since we have 52 integers, two of them must belong to a pair (x,−x). Then x^2−(−x^)2=0(mod100), so that the difference of their squares is divisible by 100.
So he creates a set, and in that set he has sets of cardinality 2. Then he says Since we
have 52 integers, two of them must belong to a pair (x,−x).
This is what I dont get. The integers can be (4, -9) or (15, 40) who says they need to be additive inverses. He just created a new set and made up some rules. I dont get it