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What is the highest power of 288 that can divide 55!?

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fauji
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What is the highest power of 288 that can divide 55!? [#permalink] New post  17 Aug 2019, 05:59
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A
B
C
D
E

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85% (hard)

Question Stats:

43% (01:51) correct 57% (02:10) wrong based on 42 sessions
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What is the highest power of 288 that can divide 55!?

A. 13
B. 11
C. 10
D. 26
E. 50

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Re: What is the highest power of 288 that can divide 55!? [#permalink] New post  17 Aug 2019, 07:16
3
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fauji wrote:
What is the highest power of 288 that can divide 55!?

A. 13
B. 11
C. 10
D. 26
E. 50


288 = 2^5*3^2

55!
Number of 2 multiples = [55/2] + [55/2^2] + [55/2^3] + [55/2^4] + [55/2^5] = 27 + 13 + 6 + 3 + 1 = 50

Number of 3 multiples = [55/3] + [55/3^2] + [55/3^3] = 18 + 6 + 2 = 26

So, 55! = 2^50*3^26*k, for some odd integer k
—> 55! = (2^5*3^2)^10*3^6*k
—> 55! = (288)^10*3^6*k

So, highest power of 288 = 10

Option C

Pls Hit kudos if you like the solution

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Re: What is the highest power of 288 that can divide 55!? [#permalink] New post  17 Aug 2019, 10:22
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kavach wrote:
Bunnel can you please teach me how to solve this question?


Hey kavach, i believe your query is to Bunuel, the Legend! Let me tag Bunuel here.

Regarding this question, let me try to explain here:

To find highest power of a 288 in 55!, we'll approach as follows:

prime factorize 288 = 2^5 * 3^2,

Now check highest power of 2^5 and 3^2 in 55!

-> check for 2 -> 27+13+6+3+1 = 50, and for 2^5 highest power becomes 10
-> check for 3 -> 18+6+2 = 26, and for 3^2 highest power becomes 13

Now, 288 is formed by 5 instances of 2 & 2 instances of 3.
Number of possible pairs of 2^5 and 3^2 = Min(10,13) => 10
Answer: Option C

Hope it is helpful!
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Re: What is the highest power of 288 that can divide 55!? [#permalink] New post  17 Aug 2019, 07:34
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Bunnel can you please teach me how to solve this question?
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Re: What is the highest power of 288 that can divide 55!? [#permalink] New post  17 Aug 2019, 10:26
Please go through following link to understand this concept:

https://gmatclub.com/forum/everything-ab ... 85592.html
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Re: What is the highest power of 288 that can divide 55!? [#permalink] New post  17 Aug 2019, 23:33
What is the question source?

Bit tough for a sub-600 question. Took me 2:01 to complete and that was without confirming that C worked for the prime factor 3...

Quote:
50% (01:08) correct

Does this include time that don't use the timer and just select an answer?
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Re: What is the highest power of 288 that can divide 55!? [#permalink] New post  17 Aug 2019, 23:52
philipssonicare wrote:
What is the question source?

Bit tough for a sub-600 question. Took me 2:01 to complete and that was without confirming that C worked for the prime factor 3...
Quote:
50% (01:08) correct

Does this include time that don't use the timer and just select an answer?



Hi philipssonicare,

It is a self made question to practice different scenarios of a concept. thought it would be helpful to share on GC.

Regarding Difficulty, once you solve few similar question, you will find it at same level. Though i am nowhere near to an expert. Please see similar topics section at the end of this page.

I am not sure about the timer doubt.

Thanks.
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Re: What is the highest power of 288 that can divide 55!? [#permalink] New post  18 Aug 2019, 00:45
fauji wrote:
What is the highest power of 288 that can divide 55!?

A. 13
B. 11
C. 10
D. 26
E. 50


\(Asked:\) What is the highest power of 288 that can divide 55!?

\(288 = 2^5*3^2\)

Power of 2 in 55! = 27+13+6+3+1 = 50
Power of 3 in 55! = 18 + 6 + 2 = 26

Power of 2^5 in 55! = [50/5] = 10
Power of 3^2 in 55! = [26/2] = 13

Power of 2^5*3^2=288 in 55! = min (10,13) = 10

IMO C
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Re: What is the highest power of 288 that can divide 55!? [#permalink] New post  03 Jan 2022, 09:53
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Re: What is the highest power of 288 that can divide 55!? [#permalink]
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