# Remainder and Factor theorem Important Formulas, Definitions, & Examples:

#### Overview:

 Topic Included: Formulas, Definitions & Exmaples. Main Topic: Quantitative Aptitude. Quantitative Aptitude Sub-topic: Number System Aptitude Notes & Questions. Questions for practice: 10 Questions & Answers with Solutions.

#### What is the Remainder Theorem:

The remainder theorem is used to find out the remainder of any expression $$f(x)$$ by dividing from a divisor $$f(c)$$. where $$f(x)$$ and $$f(c)$$ could be any linear, polynomial equation or any number.

#### Example of the Remainder theorem in real life:

If we have to distribute 5 bananas to 4 people, then one banana will be left after distribution, and this is the remainder. $$\frac{5}{4} = 1$$.

#### Basic Remainder Theorem:

We can use a simple method to find the remainder of any number or any linear equation.

Let's find the remainder of an expression $$\frac{x \times y \times z}{k}$$. When $$x, \ y, \ and \ z$$ divided by $$k$$ we get $$\frac{x_a \times y_a \times z_a}{k}$$, where $$x_a$$, $$y_a$$, and $$z_a$$ are remainders.

Here $$x_a$$ is the remainder of $$\frac{x}{k}$$, $$y_a$$ is the remainder of $$\frac{y}{k}$$ and $$z_a$$ is the remainder of $$\frac{z}{k}$$. We can understand it by taking some examples given below.

#### What is the Factor Theorem:

While finding the remainder of any expression, if we get zero as a remainder, then it means the divisor $$f(c)$$ is one of the factors of dividend $$f(x)$$. In another word, we can say if any dividend value $$f(x)$$ is completely divisible by the divisor $$f(c)$$, then there will be no remainder left or zero remainders.

Example (1): Find the remainder of $$40$$ when divided by $$10$$?

Solution: $$= \frac{40}{10} = 0$$ Here, $$0$$ is the remainder of $$\frac{40}{10}$$ or we can say there is no remainder as number 10 is one of the factors of 40.

#### Remainder Theorem Examples:

Example (2): Find the remainder of $$55$$ when divided by $$7$$?

Solution: $$= \frac{55}{7} = \frac{6}{7} = 6$$ Here, $$6$$ is the remainder of $$\frac{55}{7}$$.

Example (3): Find the remainder of $$9 \times 10 \times 11$$ when divided by $$4$$?

Solution: $$= \frac{9 \times 10 \times 11}{4} \\ = \frac{1 \times 2 \times 3}{4}$$ Here, $$1$$ is the remainder of $$\frac{9}{4}$$, $$2$$ is the remainder of $$\frac{10}{4}$$, and $$3$$ is the remainder of $$\frac{11}{4}$$

and the final remainder is $$\frac{1 \times 2 \times 3}{4} = \frac{6}{4} = 2$$

Example (4): Find the remainder of $$25 \times 28 \times 31$$ when divided by $$6$$?

Solution: $$= \frac{25 \times 28 \times 31}{6} \\ = \frac{1 \times 4 \times 1}{6}$$ where, $$1$$ is the remainder of $$\frac{25}{6}$$, $$4$$ is the remainder of $$\frac{28}{6}$$, and $$1$$ is the remainder of $$\frac{31}{6}$$

and the final remainder is $$\frac{1 \times 4 \times 1}{6} = \frac{4}{6} = 4$$

Example (5): Find the remainder of $$43 \times 46 \times 48$$ when divided by $$11$$?

Solution: $$= \frac{43 \times 46 \times 48}{11} \\ = \frac{10 \times 2 \times 4}{11}$$ where, $$10$$ is the remainder of $$\frac{43}{11}$$, $$2$$ is the remainder of $$\frac{46}{11}$$, and $$4$$ is the remainder of $$\frac{48}{11}$$

and the final remainder is $$\frac{10 \times 2 \times 4}{11} = \frac{80}{11} = 3$$

Example (6): Find the remainder of $$x^3$$ when divided by $$x$$?

Solution: $$= \frac{x^3}{x} = 0$$ Here, $$0$$ is the remainder of $$\frac{x^3}{x}$$ or we can say there is no remainder as $$x$$ is one of the factors of $$x^3$$. We can also say that $$x^3$$ is completely divisible by $$x$$ which is why no remainder left.

Example (7): Find the remainder of $$x$$ when divided by $$2x$$?

Solution: $$= \frac{x}{2x} = \frac{1}{2} = 1$$ Here, $$1$$ is the remainder of $$\frac{x}{2x}$$.

We have discussed the polynomial remainder theorem in the next chapter.