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Math Help - PROOF-mod arith

  1. #1
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    PROOF-mod arith

    Let N=22*31 + 11*17 + 13*19. Without actually computing N, determine the parity of N, the units digits of N, and the remainder when N is divided by 7.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by anncar View Post
    Let N=22*31 + 11*17 + 13*19. Without actually computing N, determine the parity of N, the units digits of N, and the remainder when N is divided by 7.
    parity of N is even.. since, odd*odd=odd, even*odd=even, odd+odd=even and odd+even=odd..
    the unit digit is 6.. since 2*1 + 1*7 + 3*9, and 3*9=27, hence, you just need to add 2+7+7=16..
    for the remainder, i had a long method. you just need to express N=7p+r and by Division Algo., r is the remainder which is in that case, r=3.
    Last edited by kalagota; October 30th 2007 at 06:50 PM.
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  3. #3
    Senior Member DivideBy0's Avatar
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    Quote Originally Posted by anncar View Post
    Let N=22*31 + 11*17 + 13*19. Without actually computing N, determine the parity of N, the units digits of N, and the remainder when N is divided by 7.
    To find the parity set N mod 2:

    22 \times 31 + 11 \times 17 + 13 \times 19 \equiv 0 \times 1 + 1 \times 1 + 1 \times 1 \equiv 2 \equiv 0 \pmod 2

    Therefore it is even.

    To find the remainder when divided by 7:

    22 \times 31 + 11 \times 17 + 13 \times 19 \equiv 1 \times 3 + 4 \times 3 + 6 \times 5 \equiv 1 \times 3 + 4 \times 3 + (-1) \times (-2) \equiv 3+12+2

    \equiv 17 \equiv 3 \pmod 7

    Therefore it leaves remainder 3.
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