Question : If the 9-digit number $72 x 8431y 4$ is divisible by 36, what is the value of $(\frac{x}{y}-\frac{y}{x})$ for the smallest possible value of $y$, given that $x$ and $y$ are natural numbers?
Option 1: $1 \frac{5}{7}$
Option 2: $2 \frac{1}{10}$
Option 3: $1 \frac{2}{5}$
Option 4: $2 \frac{9}{10}$
Correct Answer: $2 \frac{1}{10}$
Solution :
If $72 x 8431y 4$ is divisible by 36,
Then, $72 x 8431y 4$ is divisible by 4 and 9.
Since it is divisible by 4, $y4$ is divisible by 4.
On putting $y=2, y4 = 24$ which is divisible by 4.
$\therefore$ The smallest possible value of $y$ is 2.
Now $72 x 8431y 4$ becomes $72 x 84312 4$
As it is divisible by 9,
$7+2+x+8+4+3+1+2+4=31+x$ is divisible by 9
⇒ $x=5$
$\therefore x=5, y=2$
$(\frac{x}{y}-\frac{y}{x})=(\frac{5}{2}-\frac{2}{5})=\frac{25-4}{10}=\frac{21}{10}=2\frac{1}{10}$
Hence, the correct answer is $2\frac{1}{10}$.
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