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Math Help - (10n+3,5n+2)=1

  1. #1
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    (10n+3,5n+2)=1

    Prove that for any number n (10n+3,5n+2)=1

    The way I approached it was by linear combinations

    d=m(10n+3)+q(5n+2)
    =10mn+3m+5qn+2q
    =5n(2m+q)+3m+2q

    At this point I get stuck. I kind of wish there were some way of saying since it was a linear combination of 5,3, and 2 the gcd of the three must be one.
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  2. #2
    Moo
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    Hello,

    Just note that since 10=5x2, you may want to have a look at (10n+3)-2(5n+2) that's a common method : find a linear combination that cancels n and the result will be a multiple of the gcd!
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  3. #3
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    Thank you so much. I didn't think of just picking a combination. I always thought I needed to deal with them in the abstract *wavy hands*
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