Question

Number of solutions of a linear diophantine two-variable equation?

Answer 1

\[ax + by = c\]\(a,b,c\) are constants. How does one know the number of positive ordered pairs \((a,b)\)?

Answer 2

@mukushla I wish you were here ='(

Answer 3

for example, \(2a + 3b = 100\)

Answer 4

Please don't give me the answer... just explain

Answer 5

@Callisto :\

Answer 6

\[2a+3b=100\]\[2a=100-3b\]a is an integer. So, b must be even. Then, I count the numbers :\

Answer 7

From 1 - 10, there are 5. Then 11-20, 21-30..
But 100-3b ≥ 0 , so, b must be ≤ 33.333...

Answer 8

Oh yessssssssssssssssssssssssss!!!!!!!!!!!!!!!!

Answer 9

Actually \(b \le 33\) because it's an integer. Thanks!

Answer 10

:\ 33 is not included since it's not even

Answer 11

100-3b ≥ 0
why can't 'a' be negative ???

Answer 12

I want to know a more reasonable way to solve this question

Answer 13

positive ordered pairs

Answer 14

okk...

Answer 15

actually it's non-negative...hmm

Answer 16

what about 'a' ?

Answer 17

Where'd you learn that method? @Callisto

Answer 18

So the answer is 17?

Answer 19

100-3b ≥ 0
100 ≥ 3b
b ≤ 33.333

a and b are required to be non-negative integers

@ParthKohli No one taught me.. I just...think...Hmmm.. I don't know how to say it

Answer 20

It is!

Answer 21

I ran out of attempts before I could even post it. Dang

Answer 22

So I want to know a more formal/reasonable way to solve it!!

Answer 23

Me too =/

Answer 24

2a+3b=100
2a = 100-3b
Take mod 2 .
2a mod 2=0
100-3b ≡ 0 mod 2(as 2a is always divisible by 2,2a mod 2=0)
3b ≡ 100 mod 2
3b ≡0 mod 2(100 mod 2=0)
b≡1mod 2 ( because gcd(2,3)=1)
b=2k+1 (if x≡y mod z, then x=mz+y)
2a+6k+3=100
2a=97-6k
now since a and b must be positive,
b>0---->k>-1/2
a>0---->k<97/6
so, k can take only integer values from 0 to 16--->17 values.

Answer 25

Another method,
3=2*1+1
2=1*1+1
1=1*1+0
so, gcd(3,2)=1
now doing the reverse process of this :
2+1(-1) = 1
2+(3-2)(-1) =1
2(1 + 1)+ 3(-1) = 1
2(2) + 3(-1) =1
2(200) + 3(-100) =100
looks like 2a+3b = 100 ?
so, a =200 +3n (n is integer)
b= -100-2n
as, a,b >0
you get n>-200/3 ----> n>=-66.67
and n<=-50
so total you have 17 values of 'n' from -50 to -66

learned these 2 methods on facebook today :P

Answer 26

Facebook? Where?? :|

Answer 27

https://www.facebook.com/groups/elitemathcircle/
join it....

Answer 28

@ParthKohli might also be interested....

Answer 29

Thanks!!!

Answer 30

@hartnn: Where did you ask the question? ._.

Answer 31

i didn't someone else did....i saw the solution in comments.

Answer 32

@ParthKohli WTF

Answer 33

@DLS :-)