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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here. # 10! is divisible by 3x5y, where x and y are positive integer  Question banks Downloads My Bookmarks Reviews Important topics
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GRE Prep Club Legend  Joined: 07 Jun 2014
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GRE 1: Q167 V156 WE: Business Development (Energy and Utilities)
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10! is divisible by 3x5y, where x and y are positive integer [#permalink]
Expert's post 00:00

Question Stats: 62% (01:28) correct 37% (00:42) wrong based on 16 sessions
10! is divisible by $$3^x5^y$$, where x and y are positive integers.

 Quantity A Quantity B The greatest possible value for x Twice the greatest possible value for y

A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
[Reveal] Spoiler: OA

_________________

Sandy
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Try our free Online GRE Test GRE Prep Club Legend  Joined: 07 Jun 2014
Posts: 4857
GRE 1: Q167 V156 WE: Business Development (Energy and Utilities)
Followers: 105

Kudos [?]: 1783  , given: 397

Re: 10! is divisible by 3x5y, where x and y are positive integer [#permalink]
1
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Expert's post
Explanation

First, expand 10! as 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.

(Do not multiply all of those numbers together to get 3,628,800—it’s true that 3,628,800 is the value of 10!, but analysis of the prime factors of 10! is easier in the current form.)

Note that 10! is divisible by 3x5y, and the question asks for the greatest possible values of x and y, which is equivalent to asking, “What is the maximum number of times you can divide 3 and 5, respectively, out of 10! while still getting an integer answer?”

In the product 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1, only the multiples of 3 have 3 in their prime factors, and only the multiples of 5 have 5 in their prime factors. Here are all the primes contained in 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 and therefore in 10!:

10 = 5 × 2
9 = 3 × 3
8 = 2 × 2 × 2
7 = 7
6 = 2 × 3
5 = 5
4 = 2 × 2
3 = 3
2 = 2
1 = no primes

There are four 3’s and two 5’s total. The maximum values are x = 4 and y = 2. Therefore, the two quantities are equal.
_________________

Sandy
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Re: 10! is divisible by 3x5y, where x and y are positive integer [#permalink]
As the maximum values are x = 4 and y = 2, should not be A greater? Moderator  Joined: 18 Apr 2015
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Re: 10! is divisible by 3x5y, where x and y are positive integer [#permalink]
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Expert's post
$$10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1$$

$$(2*5) * (3^2) * (2^3) * 7 * (2*3) * 5 * 3 * (2^2) * 1$$

$$\frac{7 * 5^2 * 3^4 * 2^7 * 1}{3^x 5^y}$$

As you clearly see the quantity must be divided by $$3^x$$ and $$5^y$$.

In the numerator $$3^4$$ and $$5^2$$ , which means that the exponent of 3 is $$4 = x$$ and the exponent of 5 is $$2=y$$

A > B

I think also the answer should be A

Hope this helps.

Regards
_________________ GRE Instructor Joined: 10 Apr 2015
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Re: 10! is divisible by 3x5y, where x and y are positive integer [#permalink]
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Expert's post
Carcass wrote:
$$10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1$$

$$(2*5) * (3^2) * (2^3) * 7 * (2*3) * 5 * 3 * (2^2) * 1$$

$$\frac{7 * 5^2 * 3^4 * 2^7 * 1}{3^x 5^y}$$

As you clearly see the quantity must be divided by $$3^x$$ and $$5^y$$.

In the numerator $$3^4$$ and $$5^2$$ , which means that the exponent of 3 is $$4 = x$$ and the exponent of 5 is $$2=y$$

A > B

I think also the answer should be A

Hope this helps.

Regards

Be careful.
Quantity B = TWICE the greatest possible value for y

Cheers,
Brent
_________________

Brent Hanneson – Creator of greenlighttestprep.com Sign up for our free GRE Question of the Day emails Intern Joined: 20 Dec 2018
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Re: 10! is divisible by 3x5y, where x and y are positive integer [#permalink]
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Thanks to both of you, it is clear now.

I often make this kind of silly mistakes, I need to read more carefully.

Thanks again!
Moderator  Joined: 18 Apr 2015
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Kudos [?]: 1159 , given: 5501

Re: 10! is divisible by 3x5y, where x and y are positive integer [#permalink]
Expert's post Trueeeeeee

x = 4 and y= 2 but B is twice. So, y = 4

C is the answer.

Thank you Sir _________________ Re: 10! is divisible by 3x5y, where x and y are positive integer   [#permalink] 09 Mar 2019, 09:56
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