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If r, a and b are positive integers; is r a product of two consecutive

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If r, a and b are positive integers; is r a product of two consecutive [#permalink]

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New post 18 Jul 2015, 17:47
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If r, a and b are positive integers, is r a product of two consecutive odd integers?

1) \(r = 4a^2 - b^2\)
2) b = 1
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Re: If r, a and b are positive integers; is r a product of two consecutive [#permalink]

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New post 18 Jul 2015, 18:22
Mascarfi wrote:
If r, a and b are positive integers, is r a product of two consecutive odd integers?

1) \(r = 4a^2 - b^2\)
2) b = 1


2 Consecutive odd integers can be of the form : [2a+1 , 2a-1] or [2a+1, 2a+3] , with a \(\in\) integer.

Now, statement 2 is clearly not sufficient.

Per statement 1, \(r = 4a^2 - b^2\) ---> r = (2a+b)(2a-b). Thus r can be equal to product of 2 consecutive odd integers only if b =1 .

But per this statement, b can be any value (=3,5,4, etc.). Thus r = (2a+4)(2a-4) (this would give r = even and thus given question will be "no") or r = (2a+1)(2a-1) (this will give r = odd and thus the given question will be "yes").

Thus statement 1 is not sufficient.

Combining statements 1 and 2, we see r = (2a+1)(2a-1) and thus r is a product of 2 consecutive odd integers. Hence we get a definite answer by combining the statements and hence C is the correct answer.

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Re: If r, a and b are positive integers; is r a product of two consecutive [#permalink]

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New post 19 Jul 2015, 04:52
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Let the odd numbers be \((2x-1)\) and \((2x+1)\).

\(r\) must be of the form \((2x-1) (2x+1) = 4x^2 - 1\).

(i) \(r = 4a^2 - b^2\): In order for \(r\) to be of the form \((4x^2 - 1)\), \(b^2\) must be an odd number, but we have no idea what b is, so not sufficient.

(ii) \(b = 1\). clearly not sufficient.

(i) + (ii): clearly sufficient. Ans (C).
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Re: If r, a and b are positive integers; is r a product of two consecutive [#permalink]

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New post 21 Aug 2017, 14:16
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Re: If r, a and b are positive integers; is r a product of two consecutive   [#permalink] 21 Aug 2017, 14:16
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