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Math Help - Number theory, divisibility by 22

  1. #1
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    Exclamation Number theory, divisibility by 22

    Hi guys, I'm taking an advanced algebra course and this is one of the question in my assignment.

    Q: Decide which of the following integers are divisble by 22.
    a)0
    b)192544
    c)444
    d)-32516
    e)-195518


    I don't know if I can just use the long division to do it or I have to apply the divisibility by 11. Thanks for your help.
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  2. #2
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    Quote Originally Posted by suedenation View Post
    Hi guys, I'm taking an advanced algebra course and this is one of the question in my assignment.

    Q: Decide which of the following integers are divisble by 22.
    a)0
    b)192544
    c)444
    d)-32516
    e)-195518


    I don't know if I can just use the long division to do it or I have to apply the divisibility by 11. Thanks for your help.
    Note,
    gcd(2,11)=1
    Thus, it is equivalent to saying that it is divisible by 2 and by 11.
    Note all are even.
    Thus, check those that are divisible by 11, how?
    Check the alternate sum.
    For example,
    (b)
    4-4+5-2+9-1=11
    Which is divisible.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    Note,
    gcd(2,11)=1
    Thus, it is equivalent to saying that it is divisible by 2 and by 11.
    Note all are even.
    Thus, check those that are divisible by 11, how?
    Check the alternate sum.
    For example,
    (b)
    4-4+5-2+9-1=11
    Which is divisible.
    Just dividing by 11 also is not rocket science.

    RonL
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  4. #4
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    Quote Originally Posted by CaptainBlank View Post
    Just dividing by 11 also is not rocket science.
    ??
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  5. #5
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    ??
    he meant 22.

    Would 0 be considered divisible? (I should know this but I'm too lazy to look it up elsewhere)
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  6. #6
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    Quote Originally Posted by Quick View Post
    he meant 22.

    Would 0 be considered divisible? (I should know this but I'm too lazy to look it up elsewhere)
    Yes!
    The formal definition of divisibility is that,
    a divides b means there exists a k such as,
    b=ak
    For the integers.

    Thus, if we take k=0 then it answers they questions.
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