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Two number theory questions

  1. Aug 13, 2016 #1
    1. The problem statement, all variables and given/known data
    1. If a,b and c are natural numbers and a, b are coprime and a divides bc then prove that a divides c
    2. Prove that the lcm of a,b is ab / gcd(a,b)


    2. Relevant equations
    if a is a divisor of b then a = mb for a natural number m
    if a prime p is a divisor of ab then p is a divisor of a or a divisor of b
    3. The attempt at a solution

    1.since a is a divisor of bc so am = bc (m is a natural number)
    so a = (c)(b/m)
    so a/b = c/m
    Ok since a,b are coprime so a/b = a number that is not natural
    since a/b = c/m so c/m = a number that is not natural so c,m are coprime
    back to a = (c)(b/m)
    since a = (c)(b/m) which is a natural number, so bc must be a multiplie of m
    since c isn't a multiplie of m, b must be so
    so b is coprime with m
    now a/c = b/m
    since b is coprime with m
    a is coprime with c
    Q.E.D
    (Wanna check if my approach is correct or not)

    2.Prove that the lcm of a,b is ab/gcd(a,b)
    let a = xm , b = ym (m = gcd(a,b))

    ab/gcd(a,b) = xmym/m = xmy
    It is divisible by a and b so it satisfies being a multiplie
    here I gave up.



     
    Last edited by a moderator: Aug 13, 2016
  2. jcsd
  3. Aug 13, 2016 #2

    haruspex

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    That does not follow. 6/4 is not an integer, but 6 and 4 are not coprime.
    Consider some prime divisor of a.
     
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