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Math Help - Prove that ab divides c.

  1. #1
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    Prove that ab divides c.

    Let a|c and b|c, and (a,b)=1, prove that ab divides c.

    At first, I don't think this is right. For instance, let a=1 and b=2, then (a,b)=1. However, if c=3, then 2 can't divides 3 which implies b can't divides c......
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by rainyice View Post
    Let a|c and b|c, and (a,b)=1, prove that ab divides c.

    At first, I don't think this is right. For instance, let a=1 and b=2, then (a,b)=1. However, if c=3, then 2 can't divides 3 which implies b can't divides c......
    a|c \Rightarrow ax=c, then as \text{gcd}(a, b) = 1 and b|c we have that b|x. Thus, abx^{\prime} = c \Rightarrow ab|c.
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  3. #3
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    Quote Originally Posted by Swlabr View Post
    a|c \Rightarrow ax=c, then as \text{gcd}(a, b) = 1 and b|c we have that b|x. Thus, abx^{\prime} = c \Rightarrow ab|c.

    where does that b|x come from???
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by rainyice View Post
    where does that b|x come from???
    Because the gcd of a and b is 1, so it must divide x.

    (The b | c in my previous post was meant to be b \nmid c).
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