If 50! is divisible by 24^n, what is the maximum possible value of the

Question

If 50! is divisible by 24^n, what is the maximum possible value of the positive integer n?

A. 15
B. 16 
C. 17
D. 25
E. 47

ScottTargetTestPrep · Accepted Answer

Since 24 = 2^3 x 3, we see that there are fewer factors of 2^3 than factors of 3 in 50!, so we need to determine the number of factors of 2 in 50!. To do that, we can divide 50 by 2 successively as long as the quotient is nonzero and ignore the remainder whenever it’s nonzero.

50/2 = 25

25/2 = 12

12/2 = 6

6/2 = 3

3/2 = 1

Now, the number of factors of 2 in 50! is 25 + 12 + 6 + 3 + 1 = 47. Since 47/3 = 15 ⅔, there are 15 factors of 2^3 in 50!, and hence n = 15.

Answer: A

nick1816 · Answer

Highest power of 3 that can divide 50! is= 16+5+1=22

Highest power of  2^3  that can divide 50! is=  =15

Highest possible value of n=15

Archit3110 · Answer

50!/24^n
24=4*6
so
50!/4 = 12+3 ; 15
and 
50!/6 = 8+1; 9 
max possible value ; 15
IMO A

Sreeragc · Answer

Could you please explain what is the point in  /3, how do you know that it is the highest power that can divide?

EgmatQuantExpert · Answer

Hey  Sreeragc ,

We have posted an article on this concept.

To read the article:      Variations in Factorial Manipulation

Hope this helps you.
Regards,

EgmatQuantExpert · Answer

Solution

Given:  
 • 50! is divisible by  24^n

To find:  
 • The maximum possible value of the positive integer n

Approach and Working:  
 • 24 = 8 * 3 =  2^3  * 3
o So, the maximum number time  2^3  and 3 can divide 50! will be the maximum value of n.

Number of time  2^3  can divide 50!  
Since  2^3  is not prime, we will first find how many time 2 can divide 50!.
 • = \frac{50}{2} ] +  \frac{50}{4} ] +  \frac{50}{8} ] +  \frac{50}{16} ] +  \frac{50}{32} ]
• =25 +12 +6 + 3 +1 = 47 
Hence, 47 times 2 or  2^{47}  can divide 50!.

•  2^{47} =  (2^3)^{15}  *  2^2 
o Therefor,  8^{15}  or 15 times 8 can completely divide 50!.

Number of time 3 can divide 50!  
 • = \frac{50}{3} ] +  \frac{50}{9} ] +  \frac{50}{27} ]
• = 16 + 5 + 1 = 22
o Therefor,  3^{22}  or 22 times 8 can completely divide 50!. 
However, we only have 15 times 8.

• Hence, by multiplying 8 and 3, we can get 24 fifteen times.
Therefore, n= 15.

Hence, the correct answer is option A.
 
  Answer: A

janbol · Answer

Can anyone please provide a list of links to questions that are of similar type?

Need to practice more on this. Thanks in advance.

Bunuel · Answer

Check the following topics from our  Special Questions Directory :

12.  Trailing Zeros 
13.  Power of a number in a factorial

Hope it helps.

RahulSha · Answer

Approach- First simplify the question.
What is the maximum power of 24 that can divide 50!
What is the maximum power of (2^3 * 3) that can divide 50!

Maximum power of 2 that can divide 50!: 
2|50!|
 |25|
 |12|
 |6|
 |3|
 |1|
25+12+6+3+1=47
So, max power of 2^3 will be 47/3, which is 15 (take integer value)

Maximum power of 3 that can divide 50!:
3|50|
 |16|
 |5|
 |1|
16+5+1=22

Here 2^3 is the limiting factor with max power of 15, so we can come to a solution that 15 is the max power of 24 that can divide 50!

bumpbot · Answer

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If 50! is divisible by 24^n, what is the maximum possible value of the

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If 50! is divisible by 24^n, what is the maximum possible value of the

Prep Toolkit

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