Question

Solve in positive integers:
$5x+2y=53$.

• A. $x=1,3,5,7,9;y=24,19,14,9,4$
• B. $x=2,4,6,8,10;y=23,20,13,8,3$
• C. $x=11,13,15,17,19;y=1,3,4,6,9$
• D. None of these

Answer 1

Answer: (A)

Let $5x+2y=53$ .....(i)

On dividing by $2$, we get

$\frac{5x}{2}+y=\frac{53}{3}\Rightarrow y+2x+\frac{x}{2}=17+\frac{1}{3}\Rightarrow y+2x+\frac{x-1}{2}=17$

We have to solve for positive integers, so $x$ and $y$ are both integers.

$\Rightarrow \frac{x-1}{2}=$integer

Let the integer be $p$

$\frac{x-1}{2}=p$

$x=2p+1$ .......(ii)

Substituting $x$ in (i)

$5(2p+1)+2y=53\Rightarrow 2y=48-10p\Rightarrow y=24-5p$ ......(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

$\Rightarrow x=1,3,5,7,9$

Substituting $p$ in (iii),

$\Rightarrow y=24,19,14,9,4$

So, the complete solution set of positive integers is

$\{\begin{array}{c}x=1,3,5,7,9\\ y=24,19,14,9,4\end{array}$