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

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If n = 1 + x, where x is the product of four consecutive positive inte  [#permalink]

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29 Jan 2019, 23:56
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85% (hard)

Question Stats:

43% (01:49) correct 57% (01:27) wrong based on 68 sessions

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If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true?

I. n is odd
II. n is prime
III. n is a perfect square

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

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Re: If n = 1 + x, where x is the product of four consecutive positive inte  [#permalink]

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30 Jan 2019, 00:10
Bunuel wrote:
If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true?

I. n is odd
II. n is prime
III. n is a perfect square

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

n = 1+ x

1,2,3,4 = 24 + 1 = 25
2,3,4,6 = 144 + 1= 145

Only A satisfies the above examples

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If n = 1 + x, where x is the product of four consecutive positive inte  [#permalink]

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30 Jan 2019, 00:39
Bunuel wrote:
If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true?

I. n is odd
II. n is prime
III. n is a perfect square

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

x is the product of any four consecutive integers. Lets assume integers are 1,2,3,4
x=1*2*3*4=24
n=24+1=25

25 is odd as well as perfect square but its not prime...(Satisfying condition 1 and 3)
Since condition 2 is failing even in atleast 1 case we can safely say that 2 is not true.

So aption E --> I and III only should be correct

Also we can prove this algebraically :

n(n+1)(n+2)(n+3)+1 =$$(n^{2}+n)(n^{2}+5n+6)+1$$
=$$n^{4}+6n^{3}+11n^{2}+6n+1$$
=$$(n^{2}+3n+1)^{2}$$
--> n(n+1)(n+2)(n+3) = $$(n^{2}+3n+1)^{2} -1$$
Since we can see that in above equation that product of any four consecutive integer is one less perfect square, and since perfect squares are even number one less than even is always odd (Satisfying condition 1 and 3)
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Re: If n = 1 + x, where x is the product of four consecutive positive inte  [#permalink]

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30 Jan 2019, 02:43
Bunuel wrote:
If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true?

I. n is odd
II. n is prime
III. n is a perfect square

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

x=1,2,3,4
n= 1+24= 25
option 1 & 3 are true
IMO E
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Re: If n = 1 + x, where x is the product of four consecutive positive inte  [#permalink]

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30 Jan 2019, 10:54
KanishkM wrote:
Bunuel wrote:
If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true?

I. n is odd
II. n is prime
III. n is a perfect square

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

n = 1+ x

1,2,3,4 = 24 + 1 = 25
2,3,4,6 = 144 + 1= 145

Only A satisfies the above examples

Posted from my mobile device

2,3,4,6 are not consecutive.
you forgot 5.
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If n = 1 + x, where x is the product of four consecutive positive inte  [#permalink]

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Updated on: 31 Jan 2019, 06:30
Bunuel wrote:
If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true?

I. n is odd
II. n is prime
III. n is a perfect square

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

n is odd & a perfect square, holds for when the consecutive list is limited to (1,2,3,4) or (2,3,4,5).
n is a perfect square when we test all sequences.

Originally posted by harsheyb on 31 Jan 2019, 04:29.
Last edited by harsheyb on 31 Jan 2019, 06:30, edited 1 time in total.
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Re: If n = 1 + x, where x is the product of four consecutive positive inte  [#permalink]

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31 Jan 2019, 06:00
harsheyb wrote:
Bunuel wrote:
If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true?

I. n is odd
II. n is prime
III. n is a perfect square

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

n is odd & a perfect square, holds for when the consecutive list is limited to (1,2,3,4) or (2,3,4,5).
n is not a perfect square when we test (3,4,5,6).

When we test (3, 4, 5, 6), the value of x is 360 (3*4*5*6) and value of n = x +1 = 361, which is square of 19.

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Re: If n = 1 + x, where x is the product of four consecutive positive inte  [#permalink]

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01 Feb 2019, 18:38
Bunuel wrote:
If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true?

I. n is odd
II. n is prime
III. n is a perfect square

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

The product of 4 consecutive integers is always divisible by 4! = 24. Thus, n is always 1 more than a multiple of 24, so we know that n must be odd. So Roman numeral I is true.

However, n does not have to be prime. For example, if x = (1)(2)(3)(4) = 24, then n = 25, which is not a prime.

We see that n = 25 is a perfect square. Is n always going to be a perfect square? Let’s check some additional values of x.

If x = (2)(3)(4)(5) = 120, then n = 121, which is also a perfect square.

If x = (3)(4)(5)(6) = 360, then n = 361, which is also a perfect square (note: 361 = 19^2).

Thus, n will always be a perfect square.

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Re: If n = 1 + x, where x is the product of four consecutive positive inte   [#permalink] 01 Feb 2019, 18:38
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