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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: 56300
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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55% (hard)

Question Stats:

61% (01:40) correct 39% (02:03) wrong based on 268 sessions

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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

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Math Expert
Joined: 02 Aug 2009
Posts: 7764
Re: What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?  [#permalink]

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05 Apr 2016, 19:28
1
2
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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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: 7764
Re: What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?  [#permalink]

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31 Oct 2017, 07:01
1
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..
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
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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..
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!!
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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
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Joined: 12 Sep 2015
Posts: 3852
What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?  [#permalink]

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Updated on: 02 Apr 2018, 16:18
1
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 7

Cheers,
Brent
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Originally posted by GMATPrepNow on 02 Nov 2017, 11:09.
Last edited by GMATPrepNow on 02 Apr 2018, 16:18, edited 1 time in total.
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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!).

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

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

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Re: What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?   [#permalink] 12 Jan 2019, 07:25
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