Question

For what value of x is $\dfrac{81x+175}{512}$ divisible by $2$ ?

Answer 1

Hint: Here, we need to find out if the expression is divisible by $2$ or not. In order to be a multiple of $2$ , the expression must be of the form $2n$ , where “n” is a natural number. Taking the value of n to be $1$ , we get $\dfrac{81x+175}{512}=2$ which can be solved to get the answer of “x”.

Complete step-by-step solution:
In this problem, the expression that we are given is $\dfrac{81x+175}{512}$ . We need to find out if the expression is divisible by $2$ or not. We have learnt in our childhood that the divisibility by a number can be determined if the number which is to be divided, can be expressed as a multiple of the number which divides. For example, if we want to check if $121$ is divisible by $11$ or not, we need to express $121$ as a multiple of $11$ , or $2\left( 11 \right)$ .
Similarly, if we need to find out if the expression is divisible by $2$ or not, the expression must be a multiple of $2$ . This means,
$\Rightarrow \dfrac{81x+175}{512}=2n$ where “n” is a natural number.
Taking the least value of “n”, which is $1$ , we get,
$\begin{align}
  & \Rightarrow \dfrac{81x+175}{512}=2 \\
 & \Rightarrow 81x+175=1024 \\
 & \Rightarrow 81x=849 \\
 & \Rightarrow x=\dfrac{849}{81}=\dfrac{283}{27} \\
\end{align}$
Therefore, we can conclude that the value of “x” for which the given expression is divisible by $2$ is $\dfrac{283}{27}$ .

Note: The problem can also be solved in another way. This method requires a simple intuition. Now, for the entire fraction to be divisible by $2$ , the fraction must be reducible to a natural number, which must be a multiple of $2$ . This means that the numerator must be a minimum of twice of $512$ . This means,
 $\begin{align}
  & 81x+175=1024 \\
 & \Rightarrow 81x=849 \\
 & \Rightarrow x=\dfrac{849}{81}=\dfrac{283}{27} \\
\end{align}$ .