Question

If $a \equiv b\left( {\bmod m} \right)$and $x$ is an integer, then which of the following is incorrect?
$
  A.\left( {a + x} \right) \equiv \left( {b + x} \right)\left( {\bmod m} \right) \\
  B.\left( {a - x} \right) \equiv \left( {b - x} \right)\left( {\bmod m} \right) \\
  C.ax \equiv bx\left( {\bmod m} \right) \\
  D.\left( {a \div x} \right) \equiv \left( {b \div x} \right)\left( {\bmod m} \right) \\
$

Answer 1

Hint: In this question we use the properties of congruence i.e. when $a \equiv b$$\left( {\bmod m} \right)$and $x$ is a integer then $\left( {a + x} \right) \equiv \left( {b + x} \right)\left( {\bmod m} \right),\left( {a - x} \right) \equiv \left( {b - x} \right)\left( {\bmod m} \right),ax \equiv bx\left( {\bmod m} \right)$. Use elimination method to solve the above question.

Complete Step-by-Step solution:
According to the question , we have to find which option is incorrect. So, we start with applying the properties of congruence on every option and then eliminate those options where the property doesn't follow.
Hence , the properties of congruence are –
$1)$ If $a \equiv b\left( {\bmod m} \right)$and $c \equiv d\left( {\bmod m} \right)$, then $a + c \equiv b + d\left( {\bmod m} \right)$ and $a - c \equiv b - d\left( {\bmod m} \right)$.
$2)$ If $a \equiv b\left( {\bmod m} \right)$ and $c \equiv d\left( {\bmod m} \right)$, then $ac \equiv bd\left( {\bmod m} \right)$.
So, there are only three options that are fulfilling this property i.e. option $A,B$ and $C$.
The option $D$ did not follow the above property of congruence so it is incorrect.

Note: In such types of questions it is always advisable to remember all the properties related to that topic and use various tricks like elimination method, hit and trial method as it helps in solving questions easily and saves a lot of time.