This article further covers the ways to calculate remainders.

** ****Remainder Theorem:**

Remainder theorem states that when a polynomial function *f(x)* is divided by *(x –a)* where a is a constant, it will give us a remainder *f(a)*.

Remainder of

Let us understand this with the help of some examples.

**Example: **Find the remainder of **Solution:**

**Solution:**

Method 1:

Here, note that numerator is in the form of powers of 2 hence we have to convert the denominator also in the form of powers of 2. Here denominator 15 = 2^{4} –1

Now denominator is in the form of 2^{4}, so we can write numerator also in the form of 2^{4}.

2^{120} = (2^{4})^{30}. Now if take 2^{4} = *x*

then numerator => *x*^{30}

denominator = *x* –1.

Now by applying remainder theorem concept

*a* = 1 so remainder is *f*(*a*) = (1)^{30} = 1.

**Chinese Remainder Theorem:**

Chinese theorem is used when we have a big composite number in denominator and complex power in numerator.

Let us understand this with the help of some examples:

** ****Case 1: When remainder obtained by both numbers is equal **

** ****Example: **Find the remainder of

**Solution: **

**Step 1**:

Write the denominator as product of two coprimes

In this case 15 = 5 * 3

And dedude remainder for respective factor

**Step 2**:

**Case 2: When remainders obtained by both numbers are unequal **

** ****Example: **Find the remainder of

**Solution: **

**Step 1:**

Step 2:

Now we need values of a and b for that N is same

Try out some more questions based on this concept so as to get a good hold of this topic.

**EXERCISE:**

**Question 1.** Find the remainder of .

(1) 22

(2) -22

(3) 11

(4) 0

### Answer and Explanation

**Solution: Option 2**

Here given function is f(x) = 3×3 –3x –4, and it is divided by (x + 2) hence here a = –2, so remainder of this division if *f*(–2) = 3 × (–2)^{3} –3 × (–2) –4 = –24 + 6–4 = –22

**Question 2.** Find remainder of .

(1) 12

(2) 32

(3) 162

(4) 0

### Answer and Explanation

**Solution: Option 3**

**Step 1:**

Split 100 into two coprimes

Since 100 = 25 × 4 hence we will 1^{st} find remainder of

Step 2:

**Question 3.** Find remainder of

(1) 37

(2) 52

(3) 16

(4) 61

### Answer and Explanation

**Solution: Option 4**

Step 1 :

**Split 100 into two coprimes **

Since 100 = 25 × 4 hence we will 1^{st} find remainder of

Step 2:-

Step 3 :

We need to find value of **a** for that