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Math Expert V
Joined: 02 Sep 2009
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If r, a and b are positive integers, is r a product of two consecutive  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 58% (01:34) correct 42% (01:33) wrong based on 38 sessions

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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

Project DS Butler Data Sufficiency (DS3)

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Joined: 04 Feb 2020
Posts: 3
If r, a and b are positive integers, is r a product of two consecutive  [#permalink]

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9
A.
r = 4a^2 - b^2
r = (2a)^2 - b^2
r = (2a+b)(2a-b)

This means that r is the product of an even number minus b and that even number plus b.
If b is even, then r is the product of two even numbers while if b is odd, r is the product of two odd numbers.
Hence, INSUFFICIENT.

B.
Clearly INSUFFICIENT.

Combining A and B, we get r = (2a+1)(2a-1) which is the product of two odd numbers.
Hence, SUFFICIENT.
##### General Discussion
GMAT Club Legend  V
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If r, a and b are positive integers, is r a product of two consecutive  [#permalink]

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Bunuel 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

Project DS Butler Data Sufficiency (DS3)

Question: Is r product of two consecutive odd inetegers?

Statement 1: r = 4a^2 - b^2
@a = 1 and b = 1, r = 3 = 1*3 i.e. Answer to Question is YES
@a = 2 and b = 2, r = 12 = 3*4 i.e. Answer to Question is NO

NOT SUFFICIENT

Statement 2: b = 1

No connection between r and b hence

NOT SUFFICIENT

COmbining the two statements

@a = 1 and b = 1, r = 3 = 1*3 i.e. Answer to Question is YES
@a = 2 and b = 1, r = 15 = 3*5 i.e. Answer to Question is YES (Induction says SUFFICIENT)

ALGEBRA: r = 4a^2 - b^2 = r = 4a^2 - 1 = (2a+1)*(2a-1) which is product of two ODD integers hence

SUFFICIENT

Answetr: Option C
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Originally posted by GMATinsight on 27 Feb 2020, 00:07.
Last edited by GMATinsight on 27 Feb 2020, 02:59, edited 1 time in total.
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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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2
GMATinsight wrote:
Bunuel 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

Project DS Butler Data Sufficiency (DS3)

Question: Is r product of two consecutive odd inetegers?

Statement 1: r = 4a^2 - b^2
@a = 1 and b = 1, r = 3 = 1*3 i.e. Answer to Question is YES
@a = 2 and b = 2, r = 12 = 3*4 i.e. Answer to Question is NO

NOT SUFFICIENT

Statement 2: b = 1
@a = 1 and b = 1, r = 3 = 1*3 i.e. Answer to Question is YES
@a = 2 and b = 1, r = 15 = 3*5 i.e. Answer to Question is YES

r = 4a^2 - b^2 = r = 4a^2 - 1 = (2a+1)*(2a-1) which is product of two ODD integers hence

SUFFICIENT

Answetr: Option B

GMATinsight

Sir, Didn't you combine the first statement in statement 2 to arrive at the answer.

GMAT Club Legend  V
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 4029
Location: India
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GMAT 1: 750 Q51 V41
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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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1
1
shameekv1989 wrote:
GMATinsight wrote:
Bunuel 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

Project DS Butler Data Sufficiency (DS3)

Question: Is r product of two consecutive odd inetegers?

Statement 1: r = 4a^2 - b^2
@a = 1 and b = 1, r = 3 = 1*3 i.e. Answer to Question is YES
@a = 2 and b = 2, r = 12 = 3*4 i.e. Answer to Question is NO

NOT SUFFICIENT

Statement 2: b = 1
@a = 1 and b = 1, r = 3 = 1*3 i.e. Answer to Question is YES
@a = 2 and b = 1, r = 15 = 3*5 i.e. Answer to Question is YES

r = 4a^2 - b^2 = r = 4a^2 - 1 = (2a+1)*(2a-1) which is product of two ODD integers hence

SUFFICIENT

Answetr: Option B

GMATinsight

Sir, Didn't you combine the first statement in statement 2 to arrive at the answer.

shameekv1989 Thank you so much for pointing out. Made the correction. _________________
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GMATinsight .............(Bhoopendra Singh and Dr.Sushma Jha)
e-mail: info@GMATinsight.com l Call : +91-9999687183 / 9891333772
Online One-on-One Skype based classes l Classroom Coaching l On-demand Quant course
Check website for most affordable Quant on-Demand course 2000+ Qns (with Video explanations)
Our SUCCESS STORIES: From 620 to 760 l Q-42 to Q-49 in 40 days l 590 to 710 + Wharton l
ACCESS FREE GMAT TESTS HERE:22 FREE (FULL LENGTH) GMAT CATs LINK COLLECTION Re: If r, a and b are positive integers, is r a product of two consecutive   [#permalink] 27 Feb 2020, 03:00

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