UPSC 2016 CDS I Math Paper

Let the two-digit number be 10x+y, where x and y are the digits at the ten's and one's place respectively.
10x+y=k(x+y)....(1)
10y+x=m(x+y)......(2)
Adding (1) and (2), we have
11x+11y=(k+m)(x+y)
⇒k+m=11
⇒m=11−k
Trick :Consider two–digit numbers
12=4×(1+2)⇒k=4
Now,
21=m×(1+2)⇒m=7
Now, 7=11−4
∴m=11−k