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Math Help - Define gcd(a,b,c)

  1. #1
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    Arrow Define gcd(a,b,c)

    Question:

    Assuming that the gcd(a,b) of any two integers a and b is defined, show that the gcd(a,b,c) of any three integers is also defined, and that (a,b,c)=((a,b),c).

    My Answer so far: ( i think its totally wrong)

    Let gcd(a,b)=d so d|b, d|a.

    If gcd(a,b,c) = e exists then e|a , e|b and e|c ((1))

    We want to prove that
    gcd(a,b,c)=gcd(gcd(a,b),c)

    =gcd(d,c)
    =e

    Since (a,b)=d and e|a and e|b => e|d

    and from ((1)) e|c

    As e|d and e|c we have gcd(d,c)=e

    But i think i still need to show that (a,b,c) is definded and im not sure that i proved the statement correctly, i think i need to show that it is the greatest common divisor but i only showed that it was a divisor, im not sure. If anyone has a completly different solution that actually makes sense that would be super helpful!

    Thanks
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  2. #2
    o_O
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    Let d_1 = \gcd(a,b,c) \geq 1 and d_2 = \gcd ( \gcd(a,b), c) \geq 1.

    The idea is to show that d_1 \mid d_2 and d_2 \mid d_1 which implies d_1 = d_2.

    It shouldn't be too troublesome to see why this is the case.
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