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# If a=n! , what is value of n ?

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Joined: 02 Oct 2017
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If a=n! , what is value of n ?  [#permalink]

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08 Nov 2018, 09:49
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Difficulty:

35% (medium)

Question Stats:

59% (01:10) correct 41% (00:46) wrong based on 27 sessions

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If a=n! , what is value of n ?

i) The highest prime factor of a is 7

ii) a is divisible by 100

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Re: If a=n! , what is value of n ?  [#permalink]

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09 Nov 2018, 10:47
If a=n! , what is value of n ?

i) The highest prime factor of a is 7
It can be 7! or 8!,9!,10!
Insufficient

ii) a is divisible by 100
Clearly insufficient..
When n is equal to or greater than 10, A will be divisible by 100, as n! will contain atleast two 5s and thus atleast two 0s..

Combined only 10! Is possible
Sufficient

C
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: If a=n! , what is value of n ?  [#permalink]

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09 Nov 2018, 11:08
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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Re: If a=n! , what is value of n ? &nbs [#permalink] 09 Nov 2018, 11:08
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