Is $x\div x$ always equal to 1? If not, when is $x\div x$ not equal to 1? Explain briefly.

Hint: In this case, we should know about the rules of division and when division is not possible. In case of division by zero, the division operation will not be valid and thus the division of x with x will not be defined for x=0. Therefore. $x\div x$ will not be equal to 1 in the case of x=0.

Complete step by step answer:
No, in case of division of a number a by a number b, the result represents the number of times b should be added with itself such that the sum becomes equal to 1. The division will result in an integer when a is a multiple of b and will not be an integer when a is not a multiple of b.

In the case of $x\div x$ , the result should represent how many times x should be added with itself to obtain x or equivalently by what number should x be multiplied to obtain x. As multiplying x by one gives x for $x\ne 0$ , the result of $x\div x$ for $x\ne 0$ should be 1.

However, in the case of $x=0$, we find that no matter how many times we add x which is zero with itself, it cannot yield a non-zero number, therefore division by zero is not defined. Thus, $x\div x$ will not be defined when $x=0$.

Thus, the answer to the question will be no and $x\div x$ will not be equal to 1 when $x=0$.

Note: Division by 0 is always not defined, no matter what the numerator is. Therefore, in case of $x\div x$ when $x=0$, the result will be not defined even though the numerator itself is zero.