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

Question

(1) The highest prime factor of a is 7

(2) a is divisible by 100

chetan2u · Answer

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

fskilnik · Answer

a = n!\,\,\,\,\,\left( {n \geqslant 0\,\,\operatorname{int} } ight)

? = n

\left( 1 ight)\,\,\, \Rightarrow \,\,n \in \left\{ {7,8,9,10} ight\}\,\,\,\,\left( {11\,\,{	ext{is}}\,\,{	ext{prime}}} ight)\,\,\,;\,\,\,\,\,\left\{ \begin{gathered}
 \,{	ext{Take}}\,\,a = 7!\,\,\,\,\,\, \Rightarrow \,\,\,\,n\, = \,7\, \hfill \
 \,{	ext{Take}}\,\,a = 8!\,\,\,\,\,\, \Rightarrow \,\,\,\,n\, = \,8\,\, \hfill \ 
\end{gathered} ight.\,

\left( 2 ight)\,\,100 = {2^2} \cdot \boxed{{5^2}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,n \geqslant 10\,\,\,\,\left( {10\,\,{	ext{offers}}\,\,{	ext{the}}\,\,{	ext{2nd}}\,\,{	ext{factor}}\,\,{	ext{5}}} ight)\,\,\,\,\,;\,\,\,\,\,\,\,\,\left\{ \begin{gathered}
 \,{	ext{Take}}\,\,a = 10!\,\,\,\,\,\, \Rightarrow \,\,\,\,n\, = \,10\, \hfill \
 \,{	ext{Take}}\,\,a = 11!\,\,\,\,\,\, \Rightarrow \,\,\,\,n\, = \,11\,\, \hfill \ 
\end{gathered} ight.\,

\left( {1 + 2} ight)\,\,\left\{ \begin{gathered}
 \,n \in \left\{ {7,8,9,10} ight\}\, \hfill \
 \,n \geqslant 10 \hfill \ 
\end{gathered} ight.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,n = 10

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

Regards, 
Fabio.

Archit3110 · Answer

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

Dare Devil · Answer

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

DisciplinedPrep · Answer

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.

Archit3110 · Answer

#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

Gmat2Cracker · Answer

combining both the statements gives us 10! 
ans C

Kinshook · Answer

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

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