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# What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?

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Math Expert
Joined: 02 Sep 2009
Posts: 44398
What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111? [#permalink]

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05 Apr 2016, 07:43
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What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?

A. 2
B. 3
C. 5
D. 7
E. 11
[Reveal] Spoiler: OA

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Math Expert
Joined: 02 Aug 2009
Posts: 5727
Re: What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111? [#permalink]

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05 Apr 2016, 19:28
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Bunuel wrote:
What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?

A. 2
B. 3
C. 5
D. 7
E. 11

Hello,
$$\frac{(11! × 10! + 10! × 9!)}{111}$$...
In numerator take out 10!*9! as common term

$$\frac{10!*9!(11 × 10 + 1 )}{111}$$
$$\frac{10!*9!*111}{111}$$
$$10!*9!$$
10! has 7 as the greatest prime factor, so ans is 7
D
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Joined: 27 Dec 2016
Posts: 167
Concentration: International Business, Marketing
Re: What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111? [#permalink]

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31 Oct 2017, 06:55
Hi, could anyone please explain to me how 10!*9! has 7 as its highest prime factor? I was able to get to 10!*9! but didn't know how to proceed further.

Thank You!
Math Expert
Joined: 02 Aug 2009
Posts: 5727
Re: What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111? [#permalink]

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31 Oct 2017, 07:01
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csaluja wrote:
Hi, could anyone please explain to me how 10!*9! has 7 as its highest prime factor? I was able to get to 10!*9! but didn't know how to proceed further.

Thank You!

hi..
ill help you..
10! means product of first 10 numbers or 1*2*3*4*5*6*7*8*9*10 so this has 7 as the greatest prime factor
similarly 10!*9! = 1*2*3...10*1*2*3...*9... again &7 is the biggest prime factor
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Manager
Joined: 27 Dec 2016
Posts: 167
Concentration: International Business, Marketing
Re: What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111? [#permalink]

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31 Oct 2017, 12:18
chetan2u wrote:
csaluja wrote:
Hi, could anyone please explain to me how 10!*9! has 7 as its highest prime factor? I was able to get to 10!*9! but didn't know how to proceed further.

Thank You!

hi..
ill help you..
10! means product of first 10 numbers or 1*2*3*4*5*6*7*8*9*10 so this has 7 as the greatest prime factor
similarly 10!*9! = 1*2*3...10*1*2*3...*9... again &7 is the biggest prime factor

This was very helpful! Thank You & Kudos given!!
Intern
Joined: 22 Jun 2017
Posts: 17
Location: Italy
GPA: 3.64
What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111? [#permalink]

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02 Nov 2017, 08:00
Can someone link similar exercises? thanks
SVP
Joined: 12 Sep 2015
Posts: 2154
Location: Canada
Re: What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111? [#permalink]

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02 Nov 2017, 11:09
Expert's post
Top Contributor
Bunuel wrote:
What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?

A. 2
B. 3
C. 5
D. 7
E. 11

We are trying to find the greatest prime factor of the given value.
Since we can only find the greatest prime factor of an INTEGER, we can conclude that the numerator must be divisible by 111, so that it cancels out with the denominator.
How does this happen?

First notice that we can factor 10! out of the given numerator
That is, 11! × 10!  + 10! × 9! = 10!(11! + 9!)

Next we should recognize that there's also a 9! "hiding" within 11!
11! = (11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1)
= (11)(10)(9!)
This means (11! + 9!) = 9![(11)(10) + 1]

So, 11! × 10!  + 10! × 9! = 10!(11! + 9!)
= (10!)(9!)[(11)(10) + 1]
= (10!)(9!)[110 + 1]
= (10!)(9!)[111]

So, (11! × 10!  + 10! × 9!)/111 = (10!)(9!)[111]/111
= (10!)(9!)
= [(10)(9)(8)(7)(6)(5)(4)(3)(2)(1)][(9)(8)(7)(6)(5)(4)(3)(2)(1)]
= [(2)(5)(3)(3)(2)(2)(2)(7)(3)(2)(5)(2)(2)(3)(2)(1)][(3)(3)(2)(2)(2)(7)(3)(2)(5)(2)(2)(3)(2)(1)]

The greatest prime factor is 9

Cheers,
Brent
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Re: What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111? [#permalink]

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02 Nov 2017, 17:03
Bunuel wrote:
What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?

A. 2
B. 3
C. 5
D. 7
E. 11

We can simplify the given expression:

(11! x 10! + 10! x 9!)/111

(11 x 10 x 9! x 10! + 10! x 9!)/111

Let’s pull out the common factor of 9! x 10! from the two terms in the expression:

9! x 10! x (11 x 10 + 1)/111

9! x 10! x 111/111

9! x 10!

We see that the largest prime factor of 9! x 10! is 7 (since 11 won’t divide into either 9! or 10!).

Answer: D
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Math Expert
Joined: 02 Sep 2009
Posts: 44398
What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111? [#permalink]

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02 Nov 2017, 22:49
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Intern
Joined: 22 Jun 2017
Posts: 17
Location: Italy
GPA: 3.64
Re: What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111? [#permalink]

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03 Nov 2017, 00:33
Re: What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?   [#permalink] 03 Nov 2017, 00:33
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# What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?

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