2's and 5's

All the even numbers can be represented as $2+2+2.......$

All the odd numbers greater than 5 can be represented as $(2k+5)$ where $k ∈ W$ , which implies that all the odd numbers greater than 5 will be $5+2+2+2....$

The odd numbers below 5 are 1 and 3.

Thus, the only numbers which cannot be represented by the sum of 2's and 5's are 1 and 3.

$1+3=\boxed{4}$

this question can be solved by chicken mc nugget theorem

The Chicken McNugget Theorem (or Postage Stamp Problem or Coin Problem) states that for any two relatively prime positive integers m,n, the greatest integer that cannot be written in the form am + bn for nonnegative integers a, b is mn-m-n. A consequence of the theorem is that there are exactly(m - 1)(n - 1)/2 positive integers which cannot be expressed in the form am + bn. The proof is based on the fact that in each pair of the form (k, (m - 1)(n - 1) - k+1), exactly one element is expressible.

so highest integer is 10-2-5=3

and no of integer can't be expressed = 2

since given condition positive integers only

so possibilities are 0,1,2,3

and only solutions are 0,1,3..........giving sum 0+1+3=4