Find all integer solutions of the following equations a 15x

Find all integer solutions of the following equations:

(a) 15x + 33y = 7

(b) 19x + 99y = 3

No solutions

Since gcd(15,33)=3

So no x and y exist so that

15x+33y=7

Proof

Assume such x and y exist

So, 15x+33y=7

But, gcd(15,33)=3

So by Euclid Algorithm we can find integers, u,v so that

15u+33v=3

Multiplying this second equation by 2 and subtracting from first equation gives

15(x-2u)+33(y-2v)=1

which is not possible as this would mean gcd(15,33)=1

Hence no solutions exist

(b)

Now let us find gcd of 19 and 99

99=19*5+4 , 4= 99-5*19

19=4*5-1 , 1=4*5-19=4*99-20*19-19

Hence,

1=4*99-21*19

General solution is

1=(4+19t)*99-(21+99t)*19

HEnce multiplying by 3 gives

3=(12+57t)99-(63+297t)*19

Hence, x=-(63+297t) , y=12+57t

where t is any integer