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Math Help - Need help with number theory proof. Final Exam Review...

  1. #1
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    Need help with number theory proof. Final Exam Review...

    Does this proof seem valid?
    Can someone please guide me? I have a find exam tomorrow morning and I am trying to make sure I understand this.

    Prove: For all integers a,b, and c, if a|b and a|c then a|(b+c).

    Proof:
    Suppose that a,b, and c are any integers where a|b and a|c.

    Let b=a*k for some integer k (by definition of divisibility).

    Let c=a*p for some integer p (by definition of divisibility).

    Then,
    b+c= ak+ap (by substitution).
    Hence, ak+ap is an integer b/c integers and sums and products of integers are integers.

    Therefore, a|(b+c)= b+c=a*z for some integer z.

    So, ak+ap= az for some integer z (by substitution)

    Solve for z to prove that z is an integer:

    z=k+p.

    Therefore, a is divisible by b+c since b+c= a*z for some integer z.

    **END OF PROOF**.
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  2. #2
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    Joined
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    Quote Originally Posted by matthayzon89 View Post
    Does this proof seem valid?
    Can someone please guide me? I have a find exam tomorrow morning and I am trying to make sure I understand this.

    Prove: For all integers a,b, and c, if a|b and a|c then a|(b+c).

    Proof:
    Suppose that a,b, and c are any integers where a|b and a|c.

    Let b=a*k for some integer k (by definition of divisibility).

    Let c=a*p for some integer p (by definition of divisibility).

    Then,
    b+c= ak+ap (by substitution)
    I would stop here, and from this point continue as
    b+c= ak+ap
    = a( k+ p )
    = a z

    where z = k+p where clearly k+p is an integer, thus a | (b + c).

    I don't really follow what you did in the rest of the proof.



    Hence, ak+ap is an integer b/c integers and sums and products of integers are integers.

    Therefore, a|(b+c)= b+c=a*z for some integer z.

    So, ak+ap= az for some integer z (by substitution)

    Solve for z to prove that z is an integer:

    z=k+p.

    Therefore, a is divisible by b+c since b+c= a*z for some integer z.

    **END OF PROOF**.
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