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For every positive even integer n, the function h(n) is defined to be

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For every positive even integer n, the function h(n) is defined to be  [#permalink]

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New post 14 Dec 2018, 08:55
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For every positive even integer n, the function h(n) is defined to be the product of all even integers from 2 to n, inclusive. For instance, h(10)= 2x4x6x8x10. What is the greatest prime factor of h(94)+h(96)?

a) 89
b) 91
c) 93
d) 97
e) 99

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Re: For every positive even integer n, the function h(n) is defined to be  [#permalink]

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New post 14 Dec 2018, 10:48
h(94) = 2*4*6*....*94 = 2(1*2*3*...*47) = 2*47!
h(96) = 2*4*6*.....*96 = 2(1*2*3*...*48) = 2*48!
h(94)+h(96) = 47!(2+96) = 47!(97)
Hence the largest prime factor is 97.

D is the answer.
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Re: For every positive even integer n, the function h(n) is defined to be  [#permalink]

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New post 15 Dec 2018, 03:42
For every positive even integer n, the function h(n) is defined to be the product of all even integers from 2 to n, inclusive.
For instance, h(10)= 2x4x6x8x10.

What is the greatest prime factor of h(94)+h(96)?


\(h(94)+h(96) = 2*4*.....*94+2*4*......94*96 = (2*4*.....94)(1+96) = (2*4*.....94)*97\)

97 is a prime number so it is the greatest prime factor :)

Answer D
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Re: For every positive even integer n, the function h(n) is defined to be  [#permalink]

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New post 19 Dec 2018, 02:56
Afc0892 wrote:
h(94) = 2*4*6*....*94 = 2(1*2*3*...*47) = 2*47!
h(96) = 2*4*6*.....*96 = 2(1*2*3*...*48) = 2*48!
h(94)+h(96) = 47!(2+96) = 47!(97)
Hence the largest prime factor is 97.

D is the answer.

47! (2+96) is not equal to 47! (97). However, it is equal to 47! (98). But from this we can't sufficiently conclude that 97 is a factor of the two numbers. Liked the thinking though!!

Best, gota900
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Re: For every positive even integer n, the function h(n) is defined to be  [#permalink]

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New post 19 Dec 2018, 02:59
gota900 wrote:
Afc0892 wrote:
h(94) = 2*4*6*....*94 = 2(1*2*3*...*47) = 2*47!
h(96) = 2*4*6*.....*96 = 2(1*2*3*...*48) = 2*48!
h(94)+h(96) = 47!(2+96) = 47!(97)
Hence the largest prime factor is 97.

D is the answer.


47! (2+96) is not equal to 47! (97). However, it is equal to 47! (98). But from this we can't sufficiently conclude that 97 is a factor of the two numbers. Liked the thinking though!!

Best, gota900


Hey gota900, yes its 97*47!. Made a silly mistake while typing the answer ?

Posted from my mobile device
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Re: For every positive even integer n, the function h(n) is defined to be  [#permalink]

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New post 19 Dec 2018, 03:08
Afc0892 wrote:
gota900 wrote:
Afc0892 wrote:
h(94) = 2*4*6*....*94 = 2(1*2*3*...*47) = 2*47!
h(96) = 2*4*6*.....*96 = 2(1*2*3*...*48) = 2*48!
h(94)+h(96) = 47!(2+96) = 47!(97)
Hence the largest prime factor is 97.

D is the answer.


47! (2+96) is not equal to 47! (97). However, it is equal to 47! (98). But from this we can't sufficiently conclude that 97 is a factor of the two numbers. Liked the thinking though!!

Best, gota900


Hey gota900, yes its 97*47!. Made a silly mistake while typing the answer ?

Posted from my mobile device


I know that it is 47! (97). Thats why the OA is D. But a little mistake in the factoring yields 98 in your answer. Not 97, thats why I liked your thinking, but the very last step went wrong. Let me show you what I mean:

2*47! + 2*48! - so far, so good!

The factoring which would yield the desired outcome, however, should be this:

2*47! (1+2*48)
94! (1+96)
94! (97)
OA: D

Agreed?

Best, gota900
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Re: For every positive even integer n, the function h(n) is defined to be  [#permalink]

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New post 31 Dec 2018, 12:03
Hello!

Can someone please give an alternative solution instead of binomial theorem?

Thank you very much!
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Re: For every positive even integer n, the function h(n) is defined to be  [#permalink]

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New post 31 Dec 2018, 21:15
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jfranciscocuencag wrote:
Hello!

Can someone please give an alternative solution instead of binomial theorem?

Thank you very much!



H(94) = 2*4*6....*94
H(96) = 2*4*6....*94*96 = H(94)*96

H(94) + H(96) = H(94) + H(94)*96 = H(94) { 1 + 96 } = H(94) * 97 = 2*4*.....*94*97

Hence greatest prime factor is 97
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Re: For every positive even integer n, the function h(n) is defined to be  [#permalink]

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New post 01 Jan 2019, 08:52
This is how i have solved.

h(94) = 2*4*6*8........94 = 2^47 ( 1 * 2 * 3 * 4 ......47) .........(i)

h(96) = 2*4*6*8........96 =2^48 ( 1 * 2 * 3 * 4 ......48 ) .........(ii)

adding above two equation and taking 2^47 ( 1 * 2 * 3 * 4 ......47) common , we get
-->2^47 ( 1 * 2 * 3 * 4 ......47) (1 + \(2*48\))
--> 2^47 ( 1 * 2 * 3 * 4 ......47) (97)

Thus 97 is our answer.

Hope it helps.
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Re: For every positive even integer n, the function h(n) is defined to be   [#permalink] 01 Jan 2019, 08:52
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