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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 ___________

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

When the digits are reversed $\text{(yx)}$  the number increased by $18.$

$10x+y = 10y+x+18$

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

$\Rightarrow 9x-9y=18$

$\Rightarrow x-y=2$

$\Rightarrow \boxed{y=x+2}$

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

Now, we get all such numbers.

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

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

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