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



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apple_pi
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@ParthKohli any ideas?

ParthKohli
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Diophantine equations.
Gotta hate them.

sauravshakya
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I dont think the question is complete

ParthKohli
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Well, do you need positive integer solutions?

apple_pi
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Yep. Smallest positive

ParthKohli
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Oh yeah, how do you define smallest \(a,b,c\)?

apple_pi
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a b c are integers. no common factors

sauravshakya
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did u mean a+b+c to be smallest

apple_pi
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that also works

ParthKohli
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Oh, you need coprime a,b,c.

apple_pi
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yep

sauravshakya
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Can someone explain me what the real question is?

apple_pi
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what is the smallest amounts of 51 and 2 required to make a multiple of 84

ParthKohli
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lol, I have \(a = b = c= 0\). ;)

apple_pi
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perhaps I should have mentioned positive :D

sauravshakya
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51*2 + 2*33=84*2

apple_pi
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Oh wait...I did

ParthKohli
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@sauravshakya Nailed it.

sauravshakya
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Note: a must be even

Callisto
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smallest amount of 52 // smallest amount of 2 // small amount of 52 and 2?

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

Callisto
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That means smallest a / smallest b/ smallest c / smallest a+b ?

apple_pi
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8(51)+6(2) = 5(84)

sauravshakya
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So u want to find minimum value of a+b+c???

apple_pi
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yes

ParthKohli
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But they aren't coprime. :\

apple_pi
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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

apple_pi
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It's all good then guys :D