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Math Help - [SOLVED] Show that r|b

  1. #1
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    [SOLVED] Show that r|b

    if r belongs integer and r is a non zerosolution of x^2+ax+b=0 (where a,b belongs to integers), prove that r|b.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Deepu View Post
    if r belongs integer and r is a non zerosolution of x^2+ax+b=0 (where a,b belongs to integers), prove that r|b.
    We are told that r^2 + ar + b = 0

    \Rightarrow r^2 + ar = -b

    \Rightarrow r(r + a) = -b

    Now what can you say?

    Spoiler:
    Since r,a \in \mathbb{Z}, we have that (r + a) \in \mathbb{Z}, so that r divides the left side of the equation. But this means it must also divide the right side. Hence, r|b
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    Quote Originally Posted by Jhevon View Post
    We are told that r^2 + ar + b = 0

    \Rightarrow r^2 + ar = -b

    \Rightarrow r(r + a) = -b

    Now what can you say?

    Spoiler:
    Since r,a \in \mathbb{Z}, we have that (r + a) \in \mathbb{Z}, so that r divides the left side of the equation. But this means it must also divide the right side. Hence, r|b
    Thank you..
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Deepu View Post
    Thank you..
    the conclusion follows even more immediately if you recall that m|n \Longleftrightarrow n = mk \text{ for some }k \in \mathbb{Z}
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