## mathmath333 one year ago Find the number positive integer solutions of 4x+6y=200

1. mathmath333

\large \color{black}{\begin{align} & \normalsize \text{Find the number positive integer solutions of}\hspace{.33em}\\~\\ & 4x+6y=200 \end{align}}

2. mathmath333

I need to find quick way

3. Empty

I don't really know but you should look into the Euclidean Algorithm I think that might be a good path to start looking into.

4. anonymous

as this is Diophantine equation i agree with @Empty

5. anonymous

so lets try 2x+3y=100

6. ganeshie8

yeah both equations are same but we don't really need euclid algorithm here as finding a particular solution seems kinda easy by inspection. By inspection, (50, 0) is one particular solution

7. anonymous

but Euclid would help in finding general solution to show it's infinite solutions right ?

8. ganeshie8

euclid is for finding particular solution since we already figured that (50, 0) is a particular soltion, we can avoid euclid

9. anonymous

:3

10. mathmath333

I found out (50,0),(47,2),.....(2,32) but that took more than 2 min

11. ganeshie8

finding "one" particular solution is enough

12. anonymous

anyway i like this way to show its infinite:- |dw:1442748565442:dw|

13. ganeshie8

after you have one particular solution, try finding the "null" solution : 2x + 3y = 0 Easy to see that (-3, 2) solves above equation. Therefore the complete solution is given by : (50, 0) + t(-3, 2)

14. ganeshie8

since you want just the positive integer solutions, solve : 50 - 3t > 0 0 + 2t > 0

15. mathmath333

50/3=16.

16. mathmath333

17. ganeshie8

right, solve it simultaneously you should get an interval of "t" as solution

18. anonymous

aha i haven't note positive :O

19. ganeshie8

I think the answer should be 16

20. mathmath333

0<t<16

21. ganeshie8

nope 0 < t < 16.66 there are exactly 16 positive integers in that interval

22. mathmath333

0<t<17

23. ganeshie8

0 < t < 50/3 leave it like that

24. mathmath333

ok u r right (50,0) doesn't count answer is 16

25. ganeshie8

Yep! lets do one more example maybe ?

26. ganeshie8

Find the number of positive integer solutions to the equation 7x + 13y = 700

27. ganeshie8

If you prefer, here are the steps : 1) Find any one particular solution by inspection 2) Find the null solution 3) Write out the complete solution : particular + null

28. mathmath333

(100,0)

29. ganeshie8

Yep, keep going

30. mathmath333

how to find null soln

31. ganeshie8

as the name says, it is the solution to the equation 7x + 13y = 0

32. ganeshie8

give it a try.. it would feel awesome if you figure out a method to find the null solution on ur own..

33. mathmath333

(-13,-7)

34. ganeshie8

Very close, but no. plug them in and see if they really produce 0

35. mathmath333

(-13,7)

36. ganeshie8

Excellent! that is one null solution. Notice that any multiple of that also works, so all the null solutions are given by  t(-13, 7) where  t  belongs to the set of integers

37. mathmath333

100-13t>0 0+7t>0

38. ganeshie8

Yes, you have skipped step3 but ok..

39. ganeshie8

go ahead find the total count

40. mathmath333

what is step 3 ?

41. mathmath333

oh this one "3) Write out the complete solution : particular + null"

42. ganeshie8

Yes, I was refering to that..

43. mathmath333

44. ganeshie8

Yep! congratulations! Now you know how to solve any linear diophantine equation of form $$ax+by=c$$

45. mathmath333

cool!

46. ganeshie8

so what was your method for finding null solution ?

47. mathmath333

ax + by = 0 (-b,a)

48. ganeshie8

thats it!

49. ganeshie8

that works always! so finding null solution is trivial as you can seethe only hard part is finding a particular solution

50. anonymous

6y=200-4x 3y=100-2x $y=\frac{ 100-2x }{ 3 }$ by hit and trial when x=2 $y=\frac{ 100-4 }{ 3 }=32$ add successively 3 to the value of x x=2+3=5,y=30 x=5+3=8,y=28 x=11,y=26 .... x=47,y=2 x=50,y=0

51. anonymous

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