Welcome! Check out our free B-School Guides to learn how you compare with other applicants.
Login or Register

x^2 + y^2 divided by 5

This topic has 1 expert reply and 5 member replies
massi2884 Rising GMAT Star Default Avatar
Joined
13 Sep 2011
Posted:
95 messages
Thanked:
3 times
Target GMAT Score:
700+
GMAT Score:
710
x^2 + y^2 divided by 5 Post Wed May 16, 2012 10:23 am
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    If x and y are integers, what is the remainder when x^2 + y^2 is divided by 5?

    1) When x-y is divided by 5, the remainder is 1
    2) When x+y is divided by, the remainder is 2

    OA C

    Need free GMAT or MBA advice from an expert? Register for Beat The GMAT now and post your question in these forums!
    Shalabh's Quants Really wants to Beat The GMAT!
    Joined
    06 Apr 2012
    Posted:
    134 messages
    Followed by:
    5 members
    Thanked:
    35 times
    Post Wed May 16, 2012 10:55 am
    massi2884 wrote:
    If x and y are integers, what is the remainder when x^2 + y^2 is divided by 5?

    1) When x-y is divided by 5, the remainder is 1
    2) When x+y is divided by, the remainder is 2

    OA C
    As is clear that from Statement 1 and 2 that by alone, we cannot infer the result.

    So Lets combine.

    To make the calculations easier, assume x - y = 1 & x + y = 5k + 2. Where k is a natural no.

    This yields x = (5k+3)/2 & y = (5k+1)/2.

    So, x^2 + y^2 = [(5k+3)/2]^2 + [(5k+1)/2]^2 = 1/4.(50 k^2 + 40 K + 10); This expression is multiple of 5, hence remainder will be 0. Answer C.

    _________________
    Shalabh Jain,
    e-GMAT Instructor

    neelgandham Community Manager
    Joined
    13 May 2011
    Posted:
    1060 messages
    Followed by:
    49 members
    Thanked:
    313 times
    Test Date:
    October 15th 2012
    Target GMAT Score:
    740
    Post Wed May 16, 2012 11:01 am
    If x and y are integers, what is the remainder when x^2 + y^2 is divided by 5?

    Quote:
    1) When x-y is divided by 5, the remainder is 1
    If x = 6 and y = 5 then x^2 + y^2 = 36 + 25 and the remainder when x^2 + y^2 is divided by 5 is 1
    If x = 4 and y = 3 then x^2 + y^2 = 16 + 9 = 25 and the remainder when x^2 + y^2 is divided by 5 is 0
    Quote:
    2) When x+y is divided by, the remainder is 2
    If x = 7 and y = 5 then x^2 + y^2 = 49 + 25 and the remainder when x^2 + y^2 is divided by 5 is 3
    If x = 4 and y = 3 then x^2 + y^2 = 16 + 9 = 25 and the remainder when x^2 + y^2 is divided by 5 is 0
    Quote:
    From 1 + 2
    1) When x-y is divided by 5, the remainder is 1. Implies that x - y is of the form 5k + 1, where k is a non negative integer.
    2) When x+y is divided by, the remainder is 2. Implies that x + y is of the form 5l + 2, where l is a non negative integer.
    So, x + y = 5l + 2and x - y = 5k + 1
    Squaring and adding, we get
    x^2 + y^2 + 2xy + x^2 + y^2 - 2xy = 25*l^2 + 4 + 20*l + 25*k^2 + 1 + 10k
    2*(x^2 + y^2) = 25*l^2 + 20*l + 25*k^2 + 10k + 5
    2*(x^2 + y^2) = 5(5*l^2 + 4*l + 5*k^2 + 2k + 1)
    2*(x^2 + y^2) = 5* An integer
    So, we can say that x^2 + y^2 is a multiple of 5 and that if x^2 + y^2 is divided by 5, it leaves a remainder of 0

    Hence C

    _________________
    Anil Gandham
    Welcome to BEATtheGMAT
    My Quant Blog | Photography | Getting Started | BTG Community rules | MBA Watch
    Check out GMAT Prep Now’s online course at http://www.gmatprepnow.com/

    Thanked by: s777
    charu_mahajan Really wants to Beat The GMAT! Default Avatar
    Joined
    31 Jan 2012
    Posted:
    111 messages
    Followed by:
    8 members
    Thanked:
    15 times
    Post Wed May 16, 2012 12:09 pm
    Quote:
    1) When x-y is divided by 5, the remainder is 1
    If (x-y)/5 -> gives a reminder 1. That means that (x-y) should be some number like 6, 11, 16, 21...etc.

    6/5 -> Remainder 1
    11/5 -> Remainder 1
    16/5 -> Remainder 1

    Quote:
    2) When x+y is divided by 5, the remainder is 2
    If (x+y)/5 -> gives a reminder 2. That means that (x+y) should be some number like 7, 12, 17, 22...etc.

    7/5 -> Remainder 2
    12/5 -> Remainder 2
    17/5 -> Remainder 2

    Quote:
    From 1 + 2
    Plugging in the values

    1) if x-y = 6 (Can be 6,11,16,21...)
    x+y = 12(Can be 7,12,17,22...)
    Solving for x
    2x = 18 x=9
    if x=9, y = 3
    (x^2 + y^2)/5 = (9^2 + 3^2)/5 = (81+9)/5 = 90/5 -> Remainder =0

    2) Cross check with another set of values
    if x-y = 11 (Can be 6,11,16, 21...)
    x+y = 17 (Can be 7,12,17,22...)
    Solving for x
    2x = 28 x=14 y=3
    (14^2 + 3^2)/5 = (196 + 9)/5 = 205/5 -> Remainder =0

    Hence Together sufficient -> C[/quote]

    vikram4689 GMAT Titan
    Joined
    01 Nov 2009
    Posted:
    1325 messages
    Followed by:
    14 members
    Thanked:
    102 times
    Post Tue May 22, 2012 9:55 pm
    ..

    _________________
    Premise: If you like my post
    Conclusion : Press the Thanks Button Wink



    Last edited by vikram4689 on Tue May 22, 2012 11:05 pm; edited 1 time in total

    GMAT/MBA Expert

    Anurag@Gurome GMAT Instructor
    Joined
    02 Apr 2010
    Posted:
    3835 messages
    Followed by:
    487 members
    Thanked:
    1788 times
    GMAT Score:
    770
    Post Tue May 22, 2012 11:00 pm
    massi2884 wrote:
    If x and y are integers, what is the remainder when x^2 + y^2 is divided by 5?

    1) When x-y is divided by 5, the remainder is 1
    2) When x+y is divided by, the remainder is 2

    OA C
    (1) When x - y is divided by 5, the remainder is 1.
    x - y = 5a + 1, so x - y can be 1, 6, 11, ...
    If x = 2, y = 1, x - y = 1, then x² + y² = 5. So, remainder = 0.
    If x = 3, y = 2, x - y = 1, then x² + y² = 13. So, remainder = 3.
    No definite answer; NOT sufficient.

    (2) When x + y is divided by, the remainder is 2.
    x + y = 5b + 2, so x + y can be 2, 7, 12, ...
    If x = 1, y = 1, x + y = 2, then x² + y² = 2. So, remainder = 2.
    If x = 5, y = 2, x + y = 7, then x² + y² = 29. So, remainder = 4.
    No definite answer; NOT sufficient.

    Combining (1) and (2), x - y = 5a + 1 and x + y = 5b + 2
    (x - y)² = (5a + 1)² or x² - 2xy + y² = 25a² + 10a + 1
    (x + y)² = (5b + 2)² or x² + 2xy + y² = 25b² + 20b + 4
    Adding the 2 equations, we get
    2(x² + y²) = 5(5a² + 5b² + 2a + 4b + 1), which clearly implies that 2(x² + y²) is divisible by 5 with remainder = 0 and so x² + y² is also divisible by 5 with remainder = 0; SUFFICIENT.

    The correct answer is C.

    _________________
    Anurag Mairal, Ph.D., MBA
    GMAT Expert, Admissions and Career Guidance
    Gurome, Inc.
    1-800-566-4043 (USA)

    Join Our Facebook Groups
    GMAT with Gurome
    https://www.facebook.com/groups/272466352793633/
    Admissions with Gurome
    https://www.facebook.com/groups/461459690536574/
    Career Advising with Gurome
    https://www.facebook.com/groups/360435787349781/

    Thanked by: s777
    ronnie1985 GMAT Destroyer!
    Joined
    23 Dec 2011
    Posted:
    626 messages
    Followed by:
    9 members
    Thanked:
    29 times
    Test Date:
    June
    Target GMAT Score:
    750
    Post Wed May 23, 2012 7:38 am
    (C) QED

    _________________
    Follow your passion, Success as perceived by others shall follow you

    Thanked by: Pike

    Best Conversation Starters

    1 shibsriz@gmail.com 32 topics
    2 abhasjha 31 topics
    3 phanikpk 20 topics
    4 prernamalhotra 13 topics
    5 vipulgoyal 11 topics
    See More Top Beat The GMAT Members...

    Most Active Experts

    1 image description GMATGuruNY

    The Princeton Review Teacher

    167 posts
    2 image description Brent@GMATPrepNow

    GMAT Prep Now Teacher

    158 posts
    3 image description ceilidh.erickson

    Manhattan GMAT Teacher

    47 posts
    4 image description CriticalSquareMBA

    Critical Square

    39 posts
    5 image description lunarpower

    Manhattan GMAT Teacher

    29 posts
    See More Top Beat The GMAT Experts