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# For an odd integer n, the function f(n) is defined as the product of a

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Manager
Joined: 07 Jun 2017
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For an odd integer n, the function f(n) is defined as the product of a [#permalink]

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03 Nov 2017, 05:44
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Difficulty:

75% (hard)

Question Stats:

51% (01:15) correct 49% (01:09) wrong based on 107 sessions

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For an odd integer n, the function f(n) is defined as the product of all odd integers from 1 to n. The lowest odd prime factor f(71)-­1 lies between…

A. 3 and 10
B. 11 and 30
C. 31 and 50
D. 51 and 70
E. 71 and above

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For an odd integer n, the function f(n) is defined as the product of a [#permalink]

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03 Nov 2017, 05:49
nkmungila wrote:
For an odd integer n, the function f(n) is defined as the product of all odd integers from 1 to n. The lowest odd prime factor f(71)-­1 lies between…

A. 3 and 10
B. 11 and 30
C. 31 and 50
D. 51 and 70
E. 71 and above

So F(n) is nothing but 1*3*....*n-2*n
Now n! has all ODD numbers from 1 to n as its factors, so n!-1 will not have any ODD numbers 1 to n as its factors..
so f(n)-1 will be EVEN number and will have 2 as its factor BUT all other prime factors are ODD and will not be a factor of f(n)-1
This basically means f(n)-1 has ONLY 2 as prime factor which is less than n

Therefore f(71)-1 will be 1**3*4*.....*69*71 - 1
and it will have ODD prime factor greater than 71...
So E
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Re: For an odd integer n, the function f(n) is defined as the product of a [#permalink]

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03 Nov 2017, 07:19
chetan2u wrote:
nkmungila wrote:
For an odd integer n, the function f(n) is defined as the product of all odd integers from 1 to n. The lowest odd prime factor f(71)-­1 lies between…

A. 3 and 10
B. 11 and 30
C. 31 and 50
D. 51 and 70
E. 71 and above

So F(n) is nothing but n!, Which is 1*2*3*....*n-1*n
Now n! has all numbers from 1 to n as its factors, so n!-1 will not have any numbers 1 to n as its factors..
This basically means n!-1 is a prime number

Therefore f(71)-1 =71!-1, which will be a prime number and ofcourse much greater than 71. It will be 1*2*3*4*.....*70*71 - 1
So E

f(n) is defined as the product of all odd integers which means F(n) = 1.3.5.7..... . how come is it n! ?
I'm confused
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For an odd integer n, the function f(n) is defined as the product of a [#permalink]

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03 Nov 2017, 09:39
2
nkmungila wrote:
For an odd integer n, the function f(n) is defined as the product of all odd integers from 1 to n. The lowest odd prime factor f(71)-­1 lies between…

A. 3 and 10
B. 11 and 30
C. 31 and 50
D. 51 and 70
E. 71 and above

Hi MadaraU - the question can be solved as per below process

$$f(71)-1$$ & $$f(71)$$ are consecutive integers hence will be co-prime. for e.g 2, 3 or 50, 51 are co-prime.

therefore $$f(71)$$ and $$f(71)-1$$ will not have any common factor.

since $$f(71)$$ is the product of all odd numbers less than $$71$$(inclusive) which means all odd numbers less than $$71$$ are factors of $$f(71)$$, hence $$f(71)-1$$ will not have any odd factor less than $$71$$

Option E
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Re: For an odd integer n, the function f(n) is defined as the product of a [#permalink]

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03 Nov 2017, 20:55
chetan2u wrote:
nkmungila wrote:
For an odd integer n, the function f(n) is defined as the product of all odd integers from 1 to n. The lowest odd prime factor f(71)-­1 lies between…

A. 3 and 10
B. 11 and 30
C. 31 and 50
D. 51 and 70
E. 71 and above

So F(n) is nothing but n!, Which is 1*2*3*....*n-1*n
Now n! has all numbers from 1 to n as its factors, so n!-1 will not have any numbers 1 to n as its factors..
This basically means n!-1 is a prime number

Therefore f(71)-1 =71!-1, which will be a prime number and ofcourse much greater than 71. It will be 1*2*3*4*.....*70*71 - 1
So E

f(n) is defined as the product of all odd integers which means F(n) = 1.3.5.7..... . how come is it n! ?
I'm confused

Hi

thanks, I missed out on ODD in 'product of ODD integers", although answer will still remain same.
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Joined: 29 Nov 2016
Posts: 52
Re: For an odd integer n, the function f(n) is defined as the product of a [#permalink]

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17 Feb 2018, 01:18
Hi,. My doubt is will f(71)-1 have any prime number as it's factor apart from 2? I don't think there will be any prime number, even greater than 71 as it's factor chetan2u
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For an odd integer n, the function f(n) is defined as the product of a [#permalink]

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17 Feb 2018, 03:07
1
nkmungila wrote:
For an odd integer n, the function f(n) is defined as the product of all odd integers from 1 to n. The lowest odd prime factor f(71)-­1 lies between…

A. 3 and 10
B. 11 and 30
C. 31 and 50
D. 51 and 70
E. 71 and above

Function f(n) is defined as the product of all odd integers from 1 to n

If n=7, f(7) = 7*5*3 = 105 | f(7) -1 = 104 = $$2^2$$*13

If n=11, f(11) = 11*9*7*5*3 = 10395 | f(11) - 1 = 10395 - 1 = 10394
Prime factorizing 10394, we will get 2*5197 - a prime number

Extrapolating this we will understand that the lowest odd prime factor for
the function f(n) - 1 must be greater than n.

The lowest prime factor of f(71) - 1 will always be greater than 71(Option E)
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Posts: 5928
Re: For an odd integer n, the function f(n) is defined as the product of a [#permalink]

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17 Feb 2018, 07:12
Mudit27021988 wrote:
Hi,. My doubt is will f(71)-1 have any prime number as it's factor apart from 2? I don't think there will be any prime number, even greater than 71 as it's factor chetan2u

Hi Mudit27021988,

2 will be a factor but it can and generally should have a ODD factor > 71..
example..
5!-1 = 5*3*1-1=15-1=14... prime factors 2 and 7 > 5
7!-1 = 1*3*5*7 -1 = 104 = 2*2*2*13.. here 13 > 7
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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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Re: For an odd integer n, the function f(n) is defined as the product of a [#permalink]

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24 Feb 2018, 04:51
chetan2u.

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Re: For an odd integer n, the function f(n) is defined as the product of a [#permalink]

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12 Jun 2018, 07:08
f(n) is product of all odd integer from 1 to n.
f(n) - 1 will be even integer.

As these two are consecutive integers, there will not be any common factors between the two.So next factor for f(n)-1 after 2 will be more than 71.
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Re: For an odd integer n, the function f(n) is defined as the product of a   [#permalink] 12 Jun 2018, 07:08
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