harish1986 wrote:
If x, y, and k are positive integers, is k < 10 ?
(1) \(45! = x(10^k)\)
(2) \(y\) is the cubic root of \(1.25*(10^k)\)
\(x,y,k\,\, \ge 1\,\,\,{\rm{ints}}\,\,\,\left( * \right)\)
\(k\,\,\mathop < \limits^? \,\,10\)
\(\left( 1 \right)\,\,45! = x \cdot {10^k}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,k\,\,\, \le \,\,\,\left\lfloor {{{45} \over 5}} \right\rfloor + \left\lfloor {{{45} \over {25}}} \right\rfloor \, = 10\,\,\,\,\,\left( {**} \right)\,\,\)
\(\left( {**} \right)\) See my explanation (and notation) here:
https://gmatclub.com/forum/if-n-is-the- ... 75460.html\(\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {k{\kern 1pt} \,;\,x} \right) = \left( {10\,;\,\,{{45!} \over {{{10}^{10}}}}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\,\left( {x = {{45!} \over {{{10}^{10}}}}\,\, \ge 1\,\,{\mathop{\rm int}} \,\,\,{\rm{by}}\,\,\left( {**} \right)} \right)\,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {k\,;\,x} \right) = \left( {1\,;\,\,{{45!} \over {10}}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\, \hfill \cr} \right.\)
\(\left( 2 \right)\,\,\,y = \root {3\,} \of {{5 \over 4}\left( {{{10}^k}} \right)} \,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,{5 \over 4}\left( {{2^k} \cdot {5^k}} \right) = {2^{k - 2}} \cdot {5^{k + 1}}\,\,\,\,{\rm{positive}}\,\,{\rm{perfect}}\,\,{\rm{cube}}\,\,\,\,\,\)
\(\Rightarrow \,\,\,\,\,\,\left\{ \matrix{
\,k - 2 = {\rm{mult}}\,\,{\rm{of}}\,\,{\rm{3}} \hfill \cr
k + 1 = \,\,{\rm{mult}}\,\,{\rm{of}}\,\,3 \hfill \cr} \right.\,\,\,\,\,\,\,\,\left( {k \ge 2} \right)\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,k \ge 2\,\,\,\,\,{\rm{divided}}\,\,{\rm{by}}\,\,3\,\,\,{\rm{has}}\,\,{\rm{remainder}}\,\,2\,\)
\(\left\{ \matrix{
\,{\rm{Take}}\,\,k = 2\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,k = 2 + 3 \cdot 3\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\, \hfill \cr} \right.\)
\(\left( {1 + 2} \right)\,\,\,\,\,\left\{ \matrix{
\,k \le 10\,\,\,\,{\rm{by}}\,\,\,\,\left( 1 \right) \cap \left( {**} \right) \hfill \cr
\,k \ne 10\,\,\,{\rm{by}}\,\,\left( 2 \right) \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
_________________
Fabio Skilnik ::
GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here:
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