What is the highest power of 2 in n!?

Question

GMATBusters’ Quant Quiz Question -2

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What is the highest power of 2 in n!? 
1) The highest power of 4 in n! is 7. 
2) The highest power of 6 in n! is 6.

Archit3110 · Accepted Answer

What is the highest power of 2 in n!?
1) The highest power of 4 in n! is 7.
2) The highest power of 6 in n! is 6.

#1
 The highest power of 4 in n! is 7.
so here n has to be 16 ; as we get 16!/2 + 16!/4+16!/8+16!/16 ; 8+4+2+1 ; 15 and power of 4 will be 7 ; 
or n can be 17 ; 17!/2+17!/4+17!/8+17!/16 = 8+4+2+1 ; 15 and power of 4 will be 7
at n =18 ; 18!/2+18!/4+18!/8+18!/16 ; 9+4+2+1 ; 16 so power of 4 will be 8 ; not required
so highest power of 2 will be 15 at n = 16 & 17 sufficient

#2
The highest power of 6 in n! is 6.
6 = 2*3 ; 
powers of 2 will be higher than power of 3 in any factorial so power of 6 will be as many as the power of 3
insufficient
OPTION A ;

ashwini2k6jha · Answer

Answer D
State 1:sufficient
highest power for 4=7
that is highest power for 2=2*4^7=2^15
State:2 sufficient
highest power for 6=6
that means highest power for 2= 2*4*2^6=2^9=9

uc26 · Answer

We need the value of n!
1) We cannot estimate the value of n! We get a range of values. Not sufficient.
2) We cannot estimate the value of n! We get a range of values. Not sufficient.
1&2) Still have a range of possible values for n! Not sufficient.

Answer: E

techwave · Answer

What is the highest power of 2 in n!? 
1) The highest power of 4 in n! is 7. 
2) The highest power of 6 in n! is 6.

Ans: D

Prompt seeking n=?
thus we will no highest power.

If p is prime number then highest power in a factorials is given by
 n/p + n/(p2) + n/(p3) + ……

P=2 , we will need to find n

each statement provides p and highest power, we can substitute in above formula and calculate n.

MayankSingh · Answer

IMO A

What is the highest power of 2 in n!?
1) The highest power of 4 in n! is 7.
n= 16, 17 (Cuz Max power of 2 in 15! = 11 & 18! = 16, So max power of 4 = 5 & 8 resp.)
Max power of 2 = 15
Sufficient.

2) The highest power of 6 in n! is 6.
6 (=2X3) max power = 6
So, to get max power of 3 =6 , n! = 15.16,17
Now, Max power of 2 Corresponding to these = 11, 15, 15
Not Sufficient

Kinshook · Answer

Asked: What is the highest power of 2 in n!?

1) The highest power of 4 in n! is 7. 
Highest power of 2 in n! = 14 or 15.
Highest power of 2 in 15 = 7 + 3 + 1 = 11
Highest power of 2 in 16 = 8 + 4 + 2 + 1 = 15
There is no n! With highest power of 2 = 14
Highest power of 2 in n! = 15
SUFFICIENT

2) The highest power of 6 in n! is 6. 
Highest power of 3 in n! = 6;
If n=14; Highest power of 3 in n! = 4 + 1 = 5
If n={15,16,17}; Highest power of 3 in n! = 5 + 1 = 6
Highest power of 2 in 15 = 7 + 3 + 1 = 11
Highest power of 2 in {16,17} = 8 + 4 + 2 + 1 = 15
NOT SUFFICIENT

IMO A

tyson1990 · Answer

Lets take, Fact(n) = 1X2X3X4X5X6X7X8X9X10X11X12X13X14X15X16X17X18X19X20
considering only 2 & 3, Fact(n) = 2X3X(2X2)X(2X3)X(2X2X2)X(3X3)X2X(3X2X2)X2X3X(2X2X2X2)X(2X3X3)X(2X2) x K

From St(1), Power(4) = 7
Hence, Power(2) = 14 or 15

Fact(n) =  2 X3X( 2X2)X(2 X3)X( 2X2X2 )X(3X3)X 2 X(3X 2X2)X2 X3X (2X2X2X2) X(2X3X3)X(2X2) x K
from above we can count easily to find n =16, 17. And in all cases Power(2) = 15

Hence, St(1) is sufficient.

From St(2), Power(6) = 6
Hence, Power(2) >= 6 and Power(3) = 6

Fact(n) =  2X3X(2X2)X(2X3)X(2X2 X2)X( 3X3 )X2X( 3 X2X2)X2X 3 X(2X2X2X2)X(2X3X3)X(2X2) x K
from above we can count easily to find n =15, 16, 17.

Power(2) of 15 = 11
Power(2) of 16 or 17 = 15

Hence, St(2) is not sufficient.

yashikaaggarwal · Answer

Answer is D.

Statement one: 4 appears 7 times in n!.
=> 1*2*3*4*5*6*7*8*9*10*11*12*13*14*15*16
=> 2*3*4*5*(2*3)*7*(4*2)*9*(5*2)*11*(2*2*3)*13*(7*2)*15*(4*4)
=> 2*4*2*(4*2)*(2)*(2*2)*(2)*(4*4)
=> 4 appears 7 times in N!16

Statement two: 6 appears 6 times in n!.
=> 1*2*3*4*5*6*7*8*9*10*11*12*13*14*15*16
=> 2*3*2*2*5*(2*3)*7*(2*2*2)*3*3*(5*2)*11*(2*2*3)*13*(7*2)*3*5*(2*2*2*2)

=> 2*3*2*2*(2*3)*(2*2*2)*3*3*(2)*(2*2*3)*(2)*3*(2*2*2*2)
=> (2*3) appears 6 times
=> 6 appears 6 times.
N=16!

Hence we can count 2 in N! That is 15 times.

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urshi · Answer

Answer : E

Statement 1 : There are 7 4s in n!. This means there are 14 2s. There are chances for the highest power of 2 to 15 or 14 in n! Since, we cannot conclude on the answer Statement 1 is insufficient.

Statement 2 : the number of 6s in n! is dependent on the number of 3s . Hence, Statement 2 is insufficient to deduce the highest power of 2 in n!

Statement 1 & 2 together also doesn't provide the information if the highest power of 2 is 14 or 15.

Hence , Answer is E

Posted from my mobile device

Lipun · Answer

Edited:

St1: 4=2^2. Highest pow of X in n! where X=Y^Z is given by:  . => Highest power of 2 in n! can be 14 or 15. For n=15, max power of 2 is 11, and for n=16 or n=17, max power of 2 is 15. Therefore, the power of 14 for 2 doesn't occur in factorial of any number. Ans: 15, sufficient

St2: 6=2*3. From this, we can find out the highest power of 3, but not 2. For composite number, highest power is given by the power of the largest prime number in the prime factorization of the composite number. For n=15, n=16, n=17, maximum power of 3 is 6 whereas the powers of 2 are 11, 15 and 15 respectively. Can't determine.
not sufficient.

Ans: A

jrk23 · Answer

can any expert plz reply on this. so many variance in answer. Range concept is also correct as per me. so plz explain.

Lipun · Answer

Hi  jrk23 ,

I'm no expert, but taking the liberty to explain.

S1: max power of 4 in n! is 7. 
 4=2^2 . The maximum power of 4 in n! will be given by  . The   indicate the highest integer value less than or equal to (max power of 2 in n!)/2.
This implies the numerator can be 14 or 15. In either case, the value will be 7. Next, we need to find the values of n for which maximum power of 2 is 14 or 15.
For,
n=15, the maximum power of 2 will be 11
n=16, max power of 2 is 15

As you can see, the power of 14 doesn't occur. So, power of 2 in n! will be 15. Sufficient.

S2: The max power of 6 in n! will be equal to the max power of 3 in n!. So, max power of 3 in n! equals to 6.
For,
n=14, max power of 3 is 5.
n=15, max power of 3 is 6, max power of 2 is 11.
n=16, max power of 3 is 6, max power of 2 is 15. (we can stop checking here as we obtained a different value for max power of 2)
n=17, max power of 3 is 6, max power of 2 is 15.
Thus, we can't determine if the max power of 2 in n! is 11 or 15. Not sufficient.

Hope it helps!

Thanks
Lipun

Mo2men · Answer

Can you elaborate more in highlighted please? I do not understand why you did not consider 14.

GMATBusters · Answer

Lets us take the first few even numbers ( as 2 is present in even numbers only ) 2*4*6*8*10*12*14 now count the power of 2 in the product: 1+2+1+3+1+2+1= 11 Next even no is 16 : 2^4 so the power of 2 becomes = 11+4 = 15 observe that power of 2 jumps from 11 to 15 ( we cannot have the power of 12, 13, 0r 14) in fact, we cannot have power of 2 = 2, 5, 9 either. I hope it is clear now. Happy Learning

smmc29 · Answer

(1) 4 in n! = 7

In this case, we can check that n can be either 16 or 17 -> Sufficient

(2) 6 in n! = 6

-> 1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18

We can see that 2 and 3 combines to form 6 -> 1
6 -> 1
4 and 9 combines to form (2) 6 -> 2
12 has 1 six -> 1
15 combined with any of the previous even numbers not used will yield (1) 6 -> 1

For a total of 6.

Therefore 15! must yield 6 as the highest power of 6. BUT 16! and 17! can also yield 6 as the highest power of 6 since both 16 and 17 do not have 3 in their factors. If we multiply 15! by 16 or 16x17, the result will give us an additional of 4 to the power of 2. As a result we have 2 possible highest powers for 2.

-> INSUFFICIENT

Kimberly77 · Answer

Hello all experts, could anyone explain why is n has to be 16 here and power of 4 will be 7 ? Thanks

Or could any experts explain it in a different way as couldn't understand the logic with the methods above? Thanks for your time in advanced.

Lastly is the correct answer A or E

?

Kimberly77 · Answer

Thanks for explanation  Archit3110 , not quite sure what's the reasonging behind 16 , 17, 18 and not other numbers? Thanks

Kimberly77 · Answer

ThatDudeKnows   gmatprep   ScottTargetTestPrep   GMATinsight   BrentGMATPrepNow 
could you kindly share your insights on this question please? Thanks for your time in advanced.

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What is the highest power of 2 in n!?

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