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I need to find quick way

so lets try
2x+3y=100

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

:3

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

finding "one" particular solution is enough

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

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

50/3=16.

answer is 17

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

aha i haven't note positive :O

I think the answer should be 16

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

0 < t < 50/3
leave it like that

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

Yep! lets do one more example maybe ?

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

(100,0)

Yep, keep going

how to find null soln

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

(-13,-7)

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

(-13,7)

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

Yes, you have skipped step3 but ok..

go ahead find the total count

what is step 3 ?

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

Yes, I was refering to that..

7 is answer

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

cool!

so what was your method for finding null solution ?

ax + by = 0
(-b,a)

thats it!