Data Sufficiency

Brent@GMATPrepNow wrote:j and k are positive integers, and n = 10^j + k. Is n divisible by 15?
(1) j and k are each divisible by 3
(2) j and k are each divisible by 5

Source: www.gmatprepnow.com

$$n = {\rm{1}}{{\rm{0}}^{\rm{j}}} + k\,\,\,\,;\,\,\,\,\,j,k\,\,\, \ge 1\,\,\,\,{\rm{ints}}\,\,\,\left( * \right)$$
$${n \over {3 \cdot 5}}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}} $$
$$\left( 1 \right)\,\, \cap \,\,\,\left( * \right)\,\,\, \Rightarrow \,\,\,\left\{ \matrix{
j = 3M,\,\,M \ge 1\,\,{\mathop{\rm int}} \hfill \cr
k = 3L,\,\,L \ge 1\,\,{\mathop{\rm int}} \hfill \cr} \right.$$
$$n\,\,\, = \underbrace {{{\left( {{{10}^{M\, \ge \,1}}} \right)}^3}}_{\left\langle {100 \ldots 0} \right\rangle \,\,{\rm{not}}\,\,{\rm{div}}\,\,{\rm{by}}\,\,3} + \underbrace {3L}_{{\rm{div}}\,\,{\rm{by}}\,\,3}\,\, = \,\,{\rm{not}}\,\,{\rm{div}}\,\,{\rm{by}}\,\,\,3\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle $$
$$\left( 2 \right)\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {j,k} \right) = \left( {5,5} \right)\,\,\,\,\,\, \Rightarrow \,\,\,n = 100005\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\left( {\sum\nolimits_{{\rm{digits}}} {\,\,{\rm{and}}\,\,{\rm{final}}\,\,{\rm{digit}}} } \right) \hfill \cr
\,{\rm{Take}}\,\,\left( {j,k} \right) = \left( {5,10} \right)\,\,\,\,\,\, \Rightarrow \,\,\,n = 100010\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\left( {\sum\nolimits_{{\rm{digits}}} {} } \right) \hfill \cr} \right.$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.