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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
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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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65% (01:44) correct 35% (02:01) wrong based on 259 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: 7992
Re: What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?  [#permalink]

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

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Updated on: 02 Apr 2018, 16:18
2
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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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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19 Aug 2019, 06:54
Bunuel wrote:
What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?

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

denominator: 111
numerator: 11! × 10!  + 10! × 9! = 10!(11!+9!) = 10!(11•10•9!+9!) = 10!(9!(11•10+1)) = 10!9!111
greatest prime factor: (10!9!111)/111 = 10!9! = 7

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

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23 Aug 2019, 11:48
Hi everyone,

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

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

Basically we can factor this way: (10!*9!*(10*11+1))/111

111 cancels out with 10*11+1 and we are left with 10!*9!

Of the number in the list 7 is the higher factor.

Hence option D is the correct answer
Re: What is the greatest prime factor of (11! × 10!  + 10! × 9!)/111?   [#permalink] 23 Aug 2019, 11:48
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