push12345 wrote:
If a=n! , what is value of n ?
i) The highest prime factor of a is 7
ii) a is divisible by 100
\(a = n!\,\,\,\,\,\left( {n \geqslant 0\,\,\operatorname{int} } \right)\)
\(? = n\)
\(\left( 1 \right)\,\,\, \Rightarrow \,\,n \in \left\{ {7,8,9,10} \right\}\,\,\,\,\left( {11\,\,{\text{is}}\,\,{\text{prime}}} \right)\,\,\,;\,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,a = 7!\,\,\,\,\,\, \Rightarrow \,\,\,\,n\, = \,7\, \hfill \\
\,{\text{Take}}\,\,a = 8!\,\,\,\,\,\, \Rightarrow \,\,\,\,n\, = \,8\,\, \hfill \\
\end{gathered} \right.\,\)
\(\left( 2 \right)\,\,100 = {2^2} \cdot \boxed{{5^2}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,n \geqslant 10\,\,\,\,\left( {10\,\,{\text{offers}}\,\,{\text{the}}\,\,{\text{2nd}}\,\,{\text{factor}}\,\,{\text{5}}} \right)\,\,\,\,\,;\,\,\,\,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,a = 10!\,\,\,\,\,\, \Rightarrow \,\,\,\,n\, = \,10\, \hfill \\
\,{\text{Take}}\,\,a = 11!\,\,\,\,\,\, \Rightarrow \,\,\,\,n\, = \,11\,\, \hfill \\
\end{gathered} \right.\,\)
\(\left( {1 + 2} \right)\,\,\left\{ \begin{gathered}
\,n \in \left\{ {7,8,9,10} \right\}\, \hfill \\
\,n \geqslant 10 \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,n = 10\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
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