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If a and b are positive integers, what is the remainder...

This topic has 2 expert replies and 0 member replies

If a and b are positive integers, what is the remainder...

Post Tue Oct 31, 2017 10:08 am
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    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.

    The OA is B.

    I need help with this DS question. Please, can any expert explain it for me? Thanks.

    Need free GMAT or MBA advice from an expert? Register for Beat The GMAT now and post your question in these forums!
    Post Thu Nov 09, 2017 11:01 pm
    AAPL 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.

    The OA is B.

    I need help with this DS question. Please, can any expert explain it for me? Thanks.
    We have 9^(2a+1+b) = (3^2)^(2a+1+b) = 3^[2.(2a+1+b)] = 3^(4a+2+2b)

    Thus, we are asked to determine the remainder when 3^(4a+2+2b) is divided by 10.

    The remainder will be governed by the unit digit of 3^(4a+2+2b).

    Let's understand the power cycle of 3.

    1. 3^1 = 3; the unit digit = 3;
    2. 3^2 = 9; the unit digit = 9;
    3. 3^3 = 27; the unit digit = 7;
    4. 3^4 = 81; the unit digit = 1;

    5. 3^5 = 243; the unit digit = the unit digit of 3^(4 + 1) = 3;
    6. 3^6 = 243; the unit digit = the unit digit of 3^(4 + 2) = 9;
    7. 3^7 = 2187; the unit digit = the unit digit of 3^(4 + 3) = 7;
    8. 3^8 = 6561; the unit digit = the unit digit of 3^(4 + 4) = 1

    We see that the unit digit of the power of 3 follows a cycle of 4: 3, 9, 7, and 1.

    Thus,

    9. The unit digit of 3^(4n + 1) = the unit digit of 3^1 = 3; ignore 4n;
    10. The unit digit of 3^(4n + 2) = the unit digit of 3^2 = 9; ignore 4n;
    11. The unit digit of 3^(4n + 3) = the unit digit of 3^3 = 7; ignore 4n;
    12. The unit digit of 3^(4n + 4) = the unit digit of 3^4 = 1; ignore 4n.

    Let's switch to the question.

    Question rephrased: What is the unit digit of 3^(4a+2+2b)?

    We have 3^(4a+2+2b). The unit digit of 3^(4a+2+2b) = the unit digit of 3^(2+2b); ignore 4a.

    (1) a = 3

    Since a has no role to play, and we do not have the value of b, the statement is not sufficient,

    (2) b is odd.

    We have 3^(2+2b) = 3^2(1+b) = 3^2(1 + ODD) = 3^2(EVEN) = 3^(a multiple of 4)

    The unit digit of 3^(a multiple of 4) = the unit digit of 3^4 = 1. Sufficient.

    The correct answer: B

    Hope this helps!

    -Jay
    _________________
    Manhattan Review GMAT Prep

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    Thanked by: AAPL
    Post Mon Nov 13, 2017 3:18 pm
    AAPL 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?

    Aside: Here’s a video with tips on rephrasing the target question: http://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1100

    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

    Cheers,
    Brent

    _________________
    Brent Hanneson – Founder of GMATPrepNow.com
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