In any set of 181 square integeres prove that one can always find a subset of 19 nos, whose sum is dvsbl by 19
Please tell me what is set and subset first and the make me understand the qestn
The square integers leave only ten possible remainders when divided by 19: 0,1,4,9,16,6,17,11,7,and 5. If you
have 181 squares and put them in the boxes above, then
you can see with a bit of effort that one box will contain 19
squares because if 18 were max, then 10*18<181.
Having said this, you have your special box with at least
19 squares each of which leaves the same remainder
mod 19. Call this remainder R. The sum of these squares
leaves then, the same remainder as 19R leaves, which
is zero. So you can find 19 squares (in principle) whose sum
is divisible by 19.
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