10. Let $S$ be the smallest positive multiple of 15 , that comprises exactly $3 k$ digits with $k$ ' 0 's, $k$ '3's and $k$ ' 8 's. Find the remainder when $S$ is divided by 11.
(A) 0
(B) 3
(C) 5
(D) 6
(E) 8
10. Answer; (D)
$S$ being a multiple of 5 and 3 must end with ' 0 ' and has the sum of digits divisible by 3. Since $3+8=11$, the smallest positive $k$ such that $k \times 11$ is divisible by 3 is 3 . Thus $S=300338880$ and the remainder is
$$
0-8+8-8+3-3+0-0+3=-5 \equiv 6 \quad(\bmod 11) \text {. }
$$