Sums in a Subset

There is 213 number. As we Know that 213 divisible by 3. There is 3 way make 213 subset from T multiple by 3.

Since (3,6,...,2013) have 671 elemen and this condition satisfy any 213 elemen multiple by 3. Now we must prove 671 as maximum.

We can hoice only elemen of T

1. All of ellemen 0 mod 3,

2. All of elemen 1 mod 3

3. And So all element 2 mod 3.

If we choice element of T two from 3 above. Since the number maximum of T>213, If 213 number divisible by 3, We can change one number in subset T with number different from first number of subset such that "any" subset of T not divisible by 3.

And Now, Check 3 Choice above

1. All of 0 mod 3

2. All of 1 mod 3

Or

1. All of 2 mod 3.

And The maximum number of element T when we choice the second choice from above.

$T=\{1,4,7,...,2014\}$ consist of $\fbox{672}$ element.