Question

If 2y5 is divisible by 11, what is the value of y?

Answer 1

Given that, $2y5$ is divisible by $11$.

To find out: The value of $y$.

From the test of divisibility by $11$, we know that, a number is divisible by $11$, if the difference between the sum of digits at odd places and the sum of digits at even places is either $0$ or a multiple of $11$.

For $2y5$, sum of digits at odd places $=2+5=7$

and sum of digits at even places $=y$.

Hence, $7-y$ or $y-7$ must be either $0$ or some multiple of $11$.

$\therefore 7-y=0\Rightarrow y=7$

or $7-y=11\Rightarrow y=-4$

or $7-y=22\Rightarrow y=-15$

and so on.

or

$y-7=0\Rightarrow y=7$

or $y-7=11\Rightarrow y=18$

or $y-7=22\Rightarrow y=29$

and so on.

But, we observe that when the difference is $0$, we get $y=7$

and if it is any multiple of $11$, the value is either negative or more than one digit, which are both not possible.

Hence, the value of the difference must be $0$.

$\therefore y=7$

Hence, the value of $y$ for which $2y5$ is divisible by $11$, is $7$.