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If n = 1 + x, where x is the product of four consecutive

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If n = 1 + x, where x is the product of four consecutive [#permalink] New post 04 Jun 2005, 07:45
If n = 1 + x, where x is the product of four consecutive natural numbers, then which of the following is /are true?

(I) n is odd
(II) n and x are co-prime
(III) n is a perfect square
(IV) n is not divisible by 3
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 [#permalink] New post 04 Jun 2005, 08:40
I would say I & III is always true...

N cannot be a prime..., say X1...X4 are the natural numbers, then if X1 is 0, N=1, 1 is not a prime number!





in the above case 1/3 is not perfectly divisible by 3 but if you picked numbers you see
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 [#permalink] New post 04 Jun 2005, 08:58
II.

n and x are consecutive integers.
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 [#permalink] New post 04 Jun 2005, 09:01
Dan wrote:
II.

n and x are consecutive integers.


What do you mean? are you trying to explain what "co-prime" means?
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 [#permalink] New post 04 Jun 2005, 09:27
coprimes are like 6 and 7, or 120 and 121, where the greatest common factor is 1.

but since in n = 1 + x, n and x are consecutive integers, then they are coprime.


For the other choices:

I n could be odd (1*2*3*4) or even (2*3*4*5)
II True
III 1+(1*2*3*4) = 25 perfect square; or 1+(2*3*4*5) = 120 which is not
IV 25 not div by 3; 120 it is.
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 [#permalink] New post 04 Jun 2005, 09:37
Dan wrote:
I n could be odd (1*2*3*4) or even (2*3*4*5)
II True
III 1+(1*2*3*4) = 25 perfect square; or 1+(2*3*4*5) = 120 which is not
IV 25 not div by 3; 120 it is.


1+2*3*4*5=121 which makes III and IV true (in your reasoning)

For I, n couldn't be odd since at least 2 out of 4 consecutive integers are even, and multiplying whatever number by an even number, you get an even number, and adding 1 to an even number makes it always odd.
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Re: Uhm...what's a co-prime? [#permalink] New post 04 Jun 2005, 10:43
thearch wrote:
If n = 1 + x, where x is the product of four consecutive natural numbers, then which of the following is /are true?

(I) n is odd
(II) n and x are co-prime
(III) n is a perfect square
(IV) n is not divisible by 3


(1) true - x is always even , hence n is always odd
(2) true - n and x are consecutive integers,hence are co-prime
(3) true - x is always divisible by 24 ,and any number divisible by 24 is of the form p^2 - 1 ( where p is a prime >= 5) , so since x is of the form p^2 - 1 , n = p ^ 2 ( which is a sqaure , and is add , since p is prime)
(4) true = since n is of the for p^2 where p is prime, it cannot be divisible by 3.

HMTG.
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 [#permalink] New post 04 Jun 2005, 13:22
apparently 0 could be considered a natural number
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Re: Uhm...what's a co-prime? [#permalink] New post 04 Jun 2005, 13:29
thearch wrote:
If n = 1 + x, where x is the product of four consecutive natural numbers, then which of the following is /are true?


(I) n is odd -> true even +1 is odd
(II) n and x are co-prime -> true greatest common divisor is 1
(III) n is a perfect square -> true n = 1 +(m-2)(m-1)m(m+1) =
= (m^2-m-1)^2
(IV) n is not divisible by 3 -> true since x is divisible by 3 always
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Re: Uhm...what's a co-prime? [#permalink] New post 04 Jun 2005, 13:40
thearch wrote:
If n = 1 + x, where x is the product of four consecutive natural numbers, then which of the following is /are true?

(I) n is odd
(II) n and x are co-prime
(III) n is a perfect square
(IV) n is not divisible by 3


I think I, II and III have to be true
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 [#permalink] New post 04 Jun 2005, 14:43
Dan wrote:
coprimes are like 6 and 7, or 120 and 121, where the greatest common factor is 1.

but since in n = 1 + x, n and x are consecutive integers, then they are coprime.


For the other choices:

I n could be odd (1*2*3*4) or even (2*3*4*5)
II True
III 1+(1*2*3*4) = 25 perfect square; or 1+(2*3*4*5) = 120 which is not
IV 25 not div by 3; 120 it is.


Are 6 and 7 co-prime?? I would say NO

Lets take 21 as the number 3 and 7 are co primes because there are no other factors for 21 other than 3 and 7(excluding the number itself and 1) which are prime and factors of the number 21......

I would say I,III are the right answers
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 [#permalink] New post 04 Jun 2005, 18:39
Dan wrote:
coprimes are like 6 and 7, or 120 and 121, where the greatest common factor is 1.

but since in n = 1 + x, n and x are consecutive integers, then they are coprime.


For the other choices:

I n could be odd (1*2*3*4) or even (2*3*4*5)
II True
III 1+(1*2*3*4) = 25 perfect square; or 1+(2*3*4*5) = 120 which is not
IV 25 not div by 3; 120 it is.


1+(2*3*4*5) = 121 = 11*11 not 120
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 [#permalink] New post 04 Jun 2005, 18:41
gmat2me2 wrote:
Are 6 and 7 co-prime?? I would say NO


why not? their greatest common divisor is 1, therefore they are co-primes.
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 [#permalink] New post 04 Jun 2005, 19:03
sparky wrote:
gmat2me2 wrote:
Are 6 and 7 co-prime?? I would say NO


why not? their greatest common divisor is 1, therefore they are co-primes.


Ooops my bad....Yes they are co prime indeed.......Their GCD is 1.....
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Re: Uhm...what's a co-prime? [#permalink] New post 04 Jun 2005, 20:52
sparky wrote:

(I) n is odd -> true even +1 is odd
(II) n and x are co-prime -> true greatest common divisor is 1
(III) n is a perfect square -> true n = 1 +(m-2)(m-1)m(m+1) =
= (m^2-m-1)^2
(IV) n is not divisible by 3 -> true since x is divisible by 3 always


:b: This is how one should approach the problem.
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 [#permalink] New post 05 Jun 2005, 01:56
:punk
good show guys!!
OA is:
true
true
true
true

:-D
  [#permalink] 05 Jun 2005, 01:56
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