Kamla can complete a piece of work in 24 days. Nirmala can complete this work in 18 days. Kamla started the work alone and worked for X days. After this Nirmala alone completed the remaining work in Y days, in this way the work is completed in 20 days. If X and Y are both positive integers, then select the correct option.

As per the question, x and y sum up to 20. Assume that there are 72 units of work. Kamla does three units of work a day, and Nirmala does four. We got another equation as well: 3x+4y=72. Solving both gives us x=8, y=12. 

Let's verify each option:

$$\frac{4x}{y}+1\ is\ even\ $$

$$\frac{4x}{y}+1\ $$ = $$\frac{4*8}{12}+1\ $$ $$=\frac{8}{3}+1$$

2X + Y = 22 $$\Rightarrow 2*8+12=28$$

X - Y is an odd integer. 8-12=-4 it is even.

$$x^{2} + 1$$ is a multiple of 13

$$8^{2} + 1=65$$ is a multiple of 13.

So, only option D is correct.