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

1. apple_pi

@ParthKohli any ideas?

2. ParthKohli

Diophantine equations. Gotta hate them.

3. sauravshakya

I dont think the question is complete

4. ParthKohli

Well, do you need positive integer solutions?

5. apple_pi

Yep. Smallest positive

6. ParthKohli

Oh yeah, how do you define smallest $$a,b,c$$?

7. apple_pi

a b c are integers. no common factors

8. sauravshakya

did u mean a+b+c to be smallest

9. apple_pi

that also works

10. ParthKohli

Oh, you need coprime a,b,c.

11. apple_pi

yep

12. sauravshakya

Can someone explain me what the real question is?

13. apple_pi

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

14. ParthKohli

lol, I have $$a = b = c= 0$$. ;-)

15. apple_pi

perhaps I should have mentioned positive :D

16. sauravshakya

51*2 + 2*33=84*2

17. apple_pi

Oh wait...I did

18. ParthKohli

@sauravshakya Nailed it.

19. sauravshakya

Note: a must be even

20. Callisto

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

21. ParthKohli

$51a + 2b \equiv 0\pmod{84}$So yup, $$a$$ can't be odd.

22. Callisto

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

23. apple_pi

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

24. sauravshakya

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

25. apple_pi

yes

26. ParthKohli

But they aren't coprime. :-\

27. apple_pi

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

It's all good then guys :D

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