Solve in positive integers:
5x+2y=53.
Let 5x+2y=53 .....(i)
On dividing by 2, we get
5x2+y=533⇒y+2x+x2=17+13⇒y+2x+x−12=17
We have to solve for positive integers, so x and y are both integers.
⇒x−12=integer
Let the integer be p
x−12=p
x=2p+1 .......(ii)
Substituting x in (i)
5(2p+1)+2y=53⇒2y=48−10p⇒y=24−5p ......(iii)
We see from (ii) that x<0 for p<0 and from (iii) we see that y<0 for p>4, which is not possible as we are solving for positive integers.
So, p can be equal to 0,1,2,3,4
Substituting p in (ii), we have
⇒x=1,3,5,7,9
Substituting p in (iii),
⇒y=24,19,14,9,4
So, the complete solution set of positive integers is
{x=1,3,5,7,9y=24,19,14,9,4