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

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Bunuel
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If 50! is divisible by 24^n, what is the maximum possible value of the [#permalink] New post   10 Jun 2019, 00:35
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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
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A
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Re: If 50! is divisible by 24^n, what is the maximum possible value of the [#permalink] New post   11 Jun 2019, 20:23
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Bunuel wrote:
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


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
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Re: If 50! is divisible by 24^n, what is the maximum possible value of the [#permalink] New post   10 Jun 2019, 00:50
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Highest power of 3 that can divide 50! is= 16+5+1=22

Highest power of \(2^3\) that can divide 50! is= [(25+12+6+3+1)/3]=15

Highest possible value of n=15

Bunuel wrote:
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
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Re: If 50! is divisible by 24^n, what is the maximum possible value of the [#permalink] New post   26 Jun 2019, 12:16
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nick1816 wrote:
Highest power of 3 that can divide 50! is= 16+5+1=22

Highest power of \(2^3\) that can divide 50! is= [(25+12+6+3+1)/3]=15

Highest possible value of n=15

Bunuel wrote:
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




Could you please explain what is the point in [25+12+6+3+1]/3, how do you know that it is the highest power that can divide?
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Re: If 50! is divisible by 24^n, what is the maximum possible value of the [#permalink] New post   27 Jun 2019, 04:17
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Hey Sreeragc,

We have posted an article on this concept.

To read the article: Variations in Factorial Manipulation

Hope this helps you.
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Re: If 50! is divisible by 24^n, what is the maximum possible value of the [#permalink] New post   10 Jun 2019, 08:57
Bunuel wrote:
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


50!/24^n
24=4*6
so
50!/4 = 12+3 ; 15
and
50!/6 = 8+1; 9
max possible value ; 15
IMO A
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Re: If 50! is divisible by 24^n, what is the maximum possible value of the [#permalink] New post   27 Jun 2019, 04:27
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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
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Re: If 50! is divisible by 24^n, what is the maximum possible value of the [#permalink] New post   20 Oct 2020, 09:27
Can anyone please provide a list of links to questions that are of similar type?

Need to practice more on this. Thanks in advance.
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Re: If 50! is divisible by 24^n, what is the maximum possible value of the [#permalink] New post   20 Oct 2020, 09:38
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janbol wrote:
Can anyone please provide a list of links to questions that are of similar type?

Need to practice more on this. Thanks in advance.


Check the following topics from our Special Questions Directory:

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

Hope it helps.
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Re: If 50! is divisible by 24^n, what is the maximum possible value of the [#permalink] New post   20 Oct 2020, 19:58
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!
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Re: If 50! is divisible by 24^n, what is the maximum possible value of the [#permalink]
20 Oct 2020, 19:58
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