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apple_pi

  • 3 years ago

51a+2b=84c How to find smallest solutions for a,b,c

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  1. apple_pi
    • 3 years ago
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    @ParthKohli any ideas?

  2. ParthKohli
    • 3 years ago
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    Diophantine equations. Gotta hate them.

  3. sauravshakya
    • 3 years ago
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    I dont think the question is complete

  4. ParthKohli
    • 3 years ago
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    Well, do you need positive integer solutions?

  5. apple_pi
    • 3 years ago
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    Yep. Smallest positive

  6. ParthKohli
    • 3 years ago
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    Oh yeah, how do you define smallest \(a,b,c\)?

  7. apple_pi
    • 3 years ago
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    a b c are integers. no common factors

  8. sauravshakya
    • 3 years ago
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    did u mean a+b+c to be smallest

  9. apple_pi
    • 3 years ago
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    that also works

  10. ParthKohli
    • 3 years ago
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    Oh, you need coprime a,b,c.

  11. apple_pi
    • 3 years ago
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    yep

  12. sauravshakya
    • 3 years ago
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    Can someone explain me what the real question is?

  13. apple_pi
    • 3 years ago
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    what is the smallest amounts of 51 and 2 required to make a multiple of 84

  14. ParthKohli
    • 3 years ago
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    lol, I have \(a = b = c= 0\). ;-)

  15. apple_pi
    • 3 years ago
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    perhaps I should have mentioned positive :D

  16. sauravshakya
    • 3 years ago
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    51*2 + 2*33=84*2

  17. apple_pi
    • 3 years ago
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    Oh wait...I did

  18. ParthKohli
    • 3 years ago
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    @sauravshakya Nailed it.

  19. sauravshakya
    • 3 years ago
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    Note: a must be even

  20. Callisto
    • 3 years ago
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    smallest amount of 52 // smallest amount of 2 // small amount of 52 and 2?

  21. ParthKohli
    • 3 years ago
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    \[51a + 2b \equiv 0\pmod{84}\]So yup, \(a\) can't be odd.

  22. Callisto
    • 3 years ago
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    That means smallest a / smallest b/ smallest c / smallest a+b ?

  23. apple_pi
    • 3 years ago
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    8(51)+6(2) = 5(84)

  24. sauravshakya
    • 3 years ago
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    So u want to find minimum value of a+b+c???

  25. apple_pi
    • 3 years ago
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    yes

  26. ParthKohli
    • 3 years ago
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    But they aren't coprime. :-\

  27. apple_pi
    • 3 years ago
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    Oh whoops, then I didn't mean coprime As in like a,b,c don't have any common factors across the three of them

  28. apple_pi
    • 3 years ago
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    It's all good then guys :D

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