Question

If u form a subset of integers chosen from between 1 to 3000, such that no two integers add up to a multiple of nine, what can be the maximum numbers of elements in the subset?

Answer 1

See, We can write all the numbers from 1 to 3000 as - 9k, (9k+1), (9k+2) and so till (9k+9).

Now if we divide 3000 by 9 - Quotient will be 333.

Now as the question says - Between 1 and 3000 so exlude them :)

Thus numbers having the form -
9k+1 = 333 + 2998 - Number 1 which we have counted and had to be excluded.
9k+2 = 333 + 2999

Similarly, 9k+3, 9k+4, 9k+5, 9k+6, 9k+7, 9k+8, 9k+9 = 333 (3000 Excluded)

Now what we have to do here is to choose a subset of integers such that no two integers add up to a multiple of nine.
Hence, we cannot take 9k+9.
Also, we cannot count in these pairs -
(9k+1 and 9k+8 )
(9k+2 and 9k+7 )
(9k+3 and 9k+6 )
(9k+4 and 9k+5 ) as all will be divisible by 9.

Now we have to find the maximum number of elements in the set, for that we can calculate it like this -
Elements of the form (9k+1 or 9k+ + Elements of the form (9k+2 or 9k+7) + Elements of the form (9k+3 or 9k+6) + Elements of the form (9k+4 or 9k+5)
= 333 + 334 + 333 + 333 = 1333 :)

Answer 2

so 1333 should be the answer

Answer 3

http://www.pagalguy.com/cat/official-quant-thread-for-cat-2012-part-2-25078107/7003768

Answer 4

@Asevilla5 's solution makes sense right ?

Answer 5

so let
\(S=\left\{x: 3000>x >1\right\} \)
\(u\subseteq S \)
with condition :-
\( x=x_1\times 10^1+x_2\times 10^2+x_3\times10^3+x_4\times 10^4\)


wait lol , i thought u wanna no tow digits hehe nut integers in the subset >.<

Answer 6

what are u trying to do @BSwan

Answer 7

forget , i first missunderstood the question , i thought it like this :-
If u form a subset of integers chosen from between 1 to 3000, such that no two digits of the integer add up to a multiple of nine.

Answer 8

I tried working out this problem on paper kept messing up and found that question on the website @mathmath333

Answer 9

ohkay that looks like a more tough problem

Answer 10

mentioned

Answer 11

not really rashh bhahahaha , however nice try @Asevilla5 !
i like what you did