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# What is the largest power of 3 contained in 200!

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Manager
Joined: 07 Oct 2010
Posts: 160
What is the largest power of 3 contained in 200! [#permalink]

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24 Oct 2010, 11:29
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Question Stats:

81% (01:00) correct 19% (01:00) wrong based on 560 sessions

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What is the largest power of 3 contained in 200!

A. 88
B. 48
C. 66
D. 97
E. 39

If you think this question is really a challenge for you and you appreciate this kind of question then consider giving me some KUDOS......
[Reveal] Spoiler: OA

Last edited by Bunuel on 09 Jul 2013, 08:53, edited 2 times in total.
Renamed the topic and edited the question.
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Re: Can you take this challenge - 700+ quant [#permalink]

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24 Oct 2010, 11:36
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vyassaptarashi wrote:
Try this 700+ level question

Q. What is the largest power of 3 contained in 200!

A. 88
B. 48
C. 66
D. 97
E. 39

Send me the replies people. I will provide you the official answer and method soon.

If you think this question is really a challenge for you and you appreciate this kind of question then consider giving me some KUDOS......

Answer = $$\frac{200}{3} + \frac{200}{3^2} + \frac{200}{3^3} + \frac{200}{3^4}$$
= $$66+ 22 + 7 + 2 = 97$$

D
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Posts: 44423
Re: Can you take this challenge - 700+ quant [#permalink]

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24 Oct 2010, 11:37
Expert's post
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vyassaptarashi wrote:
Try this 700+ level question

Q. What is the largest power of 3 contained in 200!

A. 88
B. 48
C. 66
D. 97
E. 39

Send me the replies people. I will provide you the official answer and method soon.

If you think this question is really a challenge for you and you appreciate this kind of question then consider giving me some KUDOS......

So for this problem:

$$\frac{200}{3}+\frac{200}{3^2}+\frac{200}{3^3}+\frac{200}{3^4}=66+22+7+2=97$$.

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Hope it helps.
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Re: Can you take this challenge - 700+ quant [#permalink]

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24 Jul 2013, 14:18
How did you know to stop at 200/3^4 and what is the theory behind this. I have been trying all of the questions you have on the put on here, but I guess i do not understand the theory behind this. Could you elaborate?

Thanks,
AK
Math Expert
Joined: 02 Sep 2009
Posts: 44423
Re: Can you take this challenge - 700+ quant [#permalink]

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24 Jul 2013, 14:20
How did you know to stop at 200/3^4 and what is the theory behind this. I have been trying all of the questions you have on the put on here, but I guess i do not understand the theory behind this. Could you elaborate?

Thanks,
AK

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Re: What is the largest power of 3 contained in 200! [#permalink]

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09 Dec 2015, 12:05
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Expert's post
Hi All,

These types of questions don't require a complex math approach - they can be solved with a bit of 'brute force' and logic.

In real basic terms, we're asked to find all of the '3s' in 200!

We can figure out that 200/3 = 66, so we know that there are at least 66 '3s' in 200! While that answer is among the 5 choices, it seems a bit too 'easy', so let's do a bit more work and list out the first few numbers that we know have a '3' in them:

3 = 3x1
6 = 3x2
9 = 3x3

Notice how both 3 and 6 have just one 3 in them, but 9 has TWO 3s (there's an 'extra' 3 that we have to account for). This implies that there are probably other numbers that include 'extra 3s' that we have to figure out:

To find those extra 3s, we have to look at numbers that contain 'powers of 3'...

3^2 = 9
3^3 = 27
3^4 = 81

3^5 = 243, but that's too big (we're only going up to 200). Keep in mind that a multiple of 81 is also a multiple of 9 and 27, so we don't want to count any of those values more than once.

200/9 = 22, so we know that there are at least 22 extra 3s (and certainly more because of the 27 and 81). With the 66 3s that we already have, those 22 extra 3s increase the total to 88. With the other extra 3s, we'll end up with MORE than 88 3s. There's only one answer that fits that logic...

[Reveal] Spoiler:
D

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Re: What is the largest power of 3 contained in 200! [#permalink]

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26 Dec 2016, 09:57
1
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vyassaptarashi wrote:
What is the largest power of 3 contained in 200!

A. 88
B. 48
C. 66
D. 97
E. 39

If you think this question is really a challenge for you and you appreciate this kind of question then consider giving me some KUDOS......

Higest Power of 3 in 200!

=> 200/3 = 66
=> 66/3 = 22
=> 22/3 = 7
=> 7/3 = 2

So, The highest power of 3 is (D) 97

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Re: What is the largest power of 3 contained in 200! [#permalink]

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28 Feb 2017, 11:50
the power of 3 in 200! is
200/3 + 200/3^2 +200/3^3+200/3^4
66+33+7+2
97.
remember we always take the quotient in expression (200/3 and others) .
Re: What is the largest power of 3 contained in 200!   [#permalink] 28 Feb 2017, 11:50
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