Question

Find the number positive integer solutions of
4x+6y=200

Answer 1

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

Answer 2

I need to find quick way

Answer 3

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.

Answer 4

as this is Diophantine equation i agree with @Empty

Answer 5

so lets try
2x+3y=100

Answer 6

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

Answer 7

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

Answer 8

euclid is for finding particular solution

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

Answer 9

:3

Answer 10

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

Answer 11

finding "one" particular solution is enough

Answer 12

anyway i like this way to show its infinite:-

Answer 13

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)`

Answer 14

since you want just the positive integer solutions, solve :

50 - 3t > 0
0 + 2t > 0

Answer 15

50/3=16.

Answer 16

answer is 17

Answer 17

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

Answer 18

aha i haven't note positive :O

Answer 19

I think the answer should be 16

Answer 20

0<t<16

Answer 21

nope

0 < t < 16.66

there are exactly 16 positive integers in that interval

Answer 22

0<t<17

Answer 23

0 < t < 50/3

leave it like that

Answer 24

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

Answer 25

Yep! lets do one more example maybe ?

Answer 26

Find the number of positive integer solutions to the equation

7x + 13y = 700

Answer 27

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`

Answer 28

(100,0)

Answer 29

Yep, keep going

Answer 30

how to find null soln

Answer 31

as the name says, it is the solution to the equation

7x + 13y = 0

Answer 32

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

Answer 33

(-13,-7)

Answer 34

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

Answer 35

(-13,7)

Answer 36

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

Answer 37

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

Answer 38

Yes, you have skipped step3 but ok..

Answer 39

go ahead find the total count

Answer 40

what is step 3 ?

Answer 41

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

Answer 42

Yes, I was refering to that..

Answer 43

7 is answer

Answer 44

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

Answer 45

cool!

Answer 46

so what was your method for finding null solution ?

Answer 47

ax + by = 0


(-b,a)

Answer 48

thats it!

Answer 49

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

Answer 50

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

Answer 51

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