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If a and b are positive integers, what is the remainder when 9(2a+1+b)

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If a and b are positive integers, what is the remainder when 9(2a+1+b) [#permalink]

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If a and b are positive integers, what is the remainder when 9^(2a+1+b) is divided by 10?

(1) a = 3
(2) b is odd.
[Reveal] Spoiler: OA

Kudos [?]: 18 [0], given: 16

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Joined: 02 Sep 2009
Posts: 42607

Kudos [?]: 135655 [0], given: 12705

Re: If a and b are positive integers, what is the remainder when 9(2a+1+b) [#permalink]

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New post 03 May 2016, 05:59

Kudos [?]: 135655 [0], given: 12705

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Re: If a and b are positive integers, what is the remainder when 9(2a+1+b) [#permalink]

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New post 13 Nov 2017, 15:19
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happy1992 wrote:
If a and b are positive integers, what is the remainder when 9^(2a+1+b) is divided by 10?

(1) a = 3
(2) b is odd.


Target question: What is the remainder when 9^(2a+1+b) is divided by 10?
This is a great candidate for rephrasing the target question.

First recognize that this is a clever way of asking, "What is the units digit of 9^(2a+1+b)?"
Notice that 153 divided by 10 equals 15 with remainder 3
Likewise, 3218 divided by 10 equals 321 with remainder 8
And 97 divided by 10 equals 9 with remainder 7
So, we can write....
REPHRASED target question: What is the units digit of 9^(2a+1+b)?

IMPORTANT: We can RE-rephrase this target question in a way that makes it super easy to analyze the statements.
To see how, let's examine some powers of 9
9^1 = 9
9^2 = 81
9^3 = 729
9^4 = 6561
.
.
.
Notice that, when the exponent is ODD, the units digit is 9
When the exponent is EVEN, the units digit is 1
So, all we need to do is determine whether or not the exponent, (2a+1+b), is ODD or EVEN
To make things easier, we should recognize that 2a is EVEN for all integer values of a.
This means 2a+1 is ODD for all integer values of a.
So, if b is ODD, then 2a+1+b = ODD + ODD = EVEN, which means the units digit of 9^(2a+1+b) is 1
And, if b is EVEN, then 2a+1+b = ODD + EVEN = ODD, which means the units digit of 9^(2a+1+b) is 9
So, to answer the target question, all we need to know is whether b is odd or even
So,.......
RE-REPHRASED target question: Is n even or odd?

Statement 1: a = 3
This is not enough information to determine whether n is even or odd
Since we cannot answer the RE-REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: b is odd
Perfect!!
Since we can answer the RE-REPHRASED target question with certainty, statement 2 is SUFFICIENT

Answer: B

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Kudos [?]: 2750 [1], given: 364

Re: If a and b are positive integers, what is the remainder when 9(2a+1+b)   [#permalink] 13 Nov 2017, 15:19
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