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# What is the greatest prime factor of 12!11! + 11!10!

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What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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15 Dec 2014, 10:10
3
8
00:00

Difficulty:

65% (hard)

Question Stats:

51% (01:49) correct 49% (01:31) wrong based on 319 sessions

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What is the greatest prime factor of 12!11! + 11!10!?

(A) 7
(B) 11
(C) 13
(D) 17
(E) 19
Manager
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Re: What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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15 Dec 2014, 10:14
1
1
Soln:

12!11!+11!10!= 12*11*10!*11*10!+11*10!*10!=10!*10!*11(11*12+1)=10!*10!*11*133=10!*10!*11*19*7

greatest prime factor in 10! is 7 which is less than 19.
So Ans is E
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Re: What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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15 Dec 2014, 10:16
1
1
uday1409 wrote:
What is the greatest prime factor of 12!11! + 11!10!?

(A) 7
(B) 11
(C) 13
(D) 17
(E) 19

12! = 12.11.10!
12!11!+11!10! = 12.11.10!.11! +11!.10!
taking 11!.10! common we have
11!.10!(12.11+1)
11!.10!(133)
11!.10!(19.7)
as can be see the greatest prime factor is 19
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Re: What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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16 Dec 2014, 17:08
=12!11! + 11!10!
=11!(12!+10!)
=11! [ 10! (12*11 + 1)]
=11! [ 10! (132 + 1)]
=11! [ 10! (133)]

133 = 7 * 19
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Re: What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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16 Dec 2014, 17:31
why did we add i in the (12.11 +1) bracket?
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Re: What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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16 Dec 2014, 21:57
$$12!11! + 11!10!$$

$$= 12 * 11 * 10! * 11! + 11! * 10!$$

$$= 11! * 10! (12 * 11 + 1)$$

$$= 11! * 10! * 133$$

Looking at the OA & 133 in the equation above, its sure that answer has to be either 17 or 19

Unfortunately, I don't know by hard the table of 17 & 19

Trick:

Either 17*9 = 133 OR 19*7 = 133

133 >>> Not divisible by 9, but 13 - 6 = 7 >> Divisible by 7

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Re: What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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16 Dec 2014, 22:58
1
taleesh wrote:
why did we add i in the (12.11 +1) bracket?

We did not add 1. We took 10! common and 1 was left. It's the same case as this:

$$6x^2 + 2x = 2x(3x + 1)$$
If you take 2x common, the second term is taken out and only 1 is left.

Similarly, when you have 12! + 10! (which is 12*11*10*9...*1 + 10*9*8...*1), you take 10! common and are left with

10*9*8...1*(12*11 + 1)
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Re: What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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22 Nov 2016, 01:57
Here we can take 11!*10! out
N=11!*10!*132
N=11!*10!*7*19
Hence Greatest Prime Factor =19

Hence E
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Re: What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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05 Dec 2016, 12:07
12!11!+11!10!
=10!11!(12(11)+1)
=10!11!(133)
=10!11!(19)(7)

E.
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Re: What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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16 Dec 2016, 13:45
12!11!+11!10! = 11!(12!+10!)
Since 11! has only 11 as the largest prime number, we exclude it from further processing.
12!+10! = 10! (11*12+1) = 10! (133)
133 = 7 * 19
19 - is prime number that is to be the highest one, answer E is correct.
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Re: What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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16 Dec 2016, 14:07
uday1409 wrote:
What is the greatest prime factor of 12!11! + 11!10!?

(A) 7
(B) 11
(C) 13
(D) 17
(E) 19

let 10! = x ( greatest prime is 7)

12! = 12*11*x = 132x
11! = 11x

(132x)*(11x)+(11x)*x = (11x)*(133x)

133 = 19*7 thus (11x)*(133x) = (19*11*7)x^2 and since the greatest prime in x = 7 still the greatest prime in x^2 is 7 and thus greatest prime is 19......E
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Re: What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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17 Dec 2016, 19:53
Solution attached
Attachments

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Re: What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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29 Jan 2019, 16:14
Hello!

I am a bit confused here, I know that there is a rule that says the following:

Any factorial +1 is coprime of all numbers less than the factorial integer, including itself, and thus do not have any factors except 1.

So, that's why 7 couldn't be a factor?

And, that means that they must have factors greater than the factorial itself?

Kind regards!
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Re: What is the greatest prime factor of 12!11! + 11!10!  [#permalink]

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29 Jan 2019, 16:38
11!(12!+10!)
11!(12*11*10!+10!)
11!*10!(12*11+1)
11!*10!(133)
11!*10!(7*19)

The greatest prime is 19→(E)

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Re: What is the greatest prime factor of 12!11! + 11!10!   [#permalink] 29 Jan 2019, 16:38
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