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

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

(1) The highest prime factor of a is 7

(2) a is divisible by 100
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Joined: 02 Aug 2009
Posts: 8634
If a = n!, what is value of n ?  [#permalink]

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

i) The highest prime factor of a is 7
Next prime factor after 7 is 11, so n<11.
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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Re: If a = n!, what is value of n ?  [#permalink]

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09 Nov 2018, 11:08
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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 ?  [#permalink]

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15 Nov 2018, 02:47
1
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

from stmn1 : n= could be 8!,9 ! ,10!so on
not sufficient

Stmnt 2: not sufficient

Combing both stmtns : n has to be 10!...
C
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Re: If a = n!, what is value of n ?  [#permalink]

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02 Jan 2019, 02:23
1
If a = n!, what is the value of n?

(1) The highest prime factor of a is 7.
if highest prime factor is 7
then it can be any 8! OR 9! OR 10!
Insufficient

(2) a is divisible by 100
for a number to be divisible by 100 it should be a multiple of 25
it cab be 10! OR 15! or 20!... and so on since 5! has only one 5 it can be divisible by 100
Insufficient

(1) and (2) together
Highest prime factor is 7 and divisible by 100
10! is the only possibility
implies a = 10!
Sufficient

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

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02 Jan 2019, 02:27
If a = n!, what is the value of n?

Statement 1: The highest prime factor of a is 7.
Values of n that satisfy this condition are 7, 8, 9, & 10

Since there's no unique value of n, Statement 1 is NOT Sufficient

Statement 2: a is divisible by 100
Values of n that satisfy this condition are 10, 11, 12, ........

Since there's no unique value of n, Statement 2 is NOT Sufficient

Combining Statement 1 & Statement 2, we get only one unique value of n that satisfies both the conditions, i.e. n = 10. Therefore, Statement 1 & Statement 2 together is sufficient to answer the question.

Hence, the Correct Answer of C.

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

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02 Jan 2019, 15:53
push12345 wrote:
If a = n!, what is value of n ?

(1) The highest prime factor of a is 7

(2) a is divisible by 100

#1: a could be 14,35,.. in sufficient

#2:
a can be any value ; 100,200,300.. in sufficient

from 1&2::
n has to be 10! so as to be divisible by 100 and also have 7 as the highest prime factor...
IMO C
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Re: If a = n!, what is value of n ?  [#permalink]

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20 Aug 2019, 03:53
combining both the statements gives us 10!
ans C
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Re: If a = n!, what is value of n ?  [#permalink]

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20 Aug 2019, 04:38
push12345 wrote:
If a = n!, what is value of n ?

(1) The highest prime factor of a is 7

(2) a is divisible by 100

Given: a = n!

Asked: What is value of n ?

(1) The highest prime factor of a is 7
n can be 7,8,9,10
NOT SUFFICIENT

(2) a is divisible by 100
$$n \geq 10$$
NOT SUFFICIENT

Combining (1) & (2)
(1) The highest prime factor of a is 7
n can be 7,8,9,10
(2) a is divisible by 100
$$n \geq 10$$
n=10
SUFFICIENT

IMO C
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Re: If a = n!, what is value of n ?   [#permalink] 20 Aug 2019, 04:38