CAT 2015 | Question: 97

Question

When you reverse the digits of the number $13$, the number increases by $18$. How many other two digit numbers increase by $18$ when their digits reversed ___________

Answer 1

Let the two-digit numbers be $xy \Rightarrow 10x+y.$

When the digits are reversed $yx$  the number increased by $18.$

$10y+x = 10x+y+18$

$\Rightarrow 10y-y+x-10x=18$

$\Rightarrow 9y-9x=18$

$\Rightarrow y-x=2$

$\Rightarrow \boxed{y=x+2}$

All the positive two-digit numbers possible $= 10x+y = 10x+x+2 = 11x+2$

Now, we get all such numbers.

• $x=1 \Rightarrow 13 \longrightarrow 31$
• $x=2 \Rightarrow 24 \longrightarrow 42$
• $x=3 \Rightarrow 35 \longrightarrow 53$
• $x=4 \Rightarrow 46 \longrightarrow 64$
• $x=5 \Rightarrow 57 \longrightarrow 75$
• $x=6 \Rightarrow 68 \longrightarrow 86$
• $x=7 \Rightarrow 79 \longrightarrow 97$
• $x=8 \Rightarrow 90 \longrightarrow 09 (90+18=108)$ (Not possible)

$\therefore$ The number of other two-digit numbers is $6.$