(i) If ‘a_{1}’ is divided by ‘n’, the remainder is ‘r_{1}’ and if ‘a_{2}’ is divided by ‘n’, the remainder is r_{2}. Then,
If a_{1}+a_{2} is divided by n, the remainder will be r_{1} + r_{2.}
If a_{1}  a_{2} is divided by n, the remainder will be r_{1}  r_{2.}
If a_{1} × a_{2} is divided by n, the remainder will be r_{1} × r_{2.}
By definition, remainder cannot be negative. But in certain cases, you can assume that for your convenience. But a negative remainder in real sense means that you need to add the divisor in the negative remainder to find the real remainder.
Example:
If 21 is divided by 5, the remainder is 1 and if 12 is divided by 5, the remainder is 2. Then, if (21 + 12 = 33) is divided by 5, the remainder will be (1 + 2 = 3).
If (21  12 = 9) is divided by 5, the remainder will be 1  2 =  1.
But if the divisor is 5,  1 is nothing but 4 (9 = 5 × 1 + 4)
So, if 9 is divided by 5, the remainder is 4 and 9 can be written as 9 = 5 × 2  1.
So here  1 is the remainder. So  1 is equivalent to 4 if the divisor is 5. Similarly  2 is equivalent to 3.
If (21 × 12 = 252) is divisible by 5, the remainder will be (1 × 2 = 2).
(ii) If two numbers ‘a_{1}’ and ‘a_{2}‘ are exactly divisible by n. Then their sum, difference and product is also exactly divisible by n.
i.e., If ‘a_{1}’ and ‘a_{2}’ are divisible by n, then
a_{1} + a_{2} is also divisible by n
a_{1}  a_{2} is also divisible by n
a_{1} × a_{2} is also divisible by n.
Example: 12 is divisible by 3 and 21 is also divisible by 3
Sol. So, 12 + 21 = 33, 12  21 =  9 and 12 × 21 = 252 all are divisible by 3.
Example 1: What is the remainder if 7^{25} is divided by 6?
Solution: If 7 is divided by 6, the remainder is 1. So if 7^{25} is divided by 6, the remainder is 1 (because 7^{25} = 7 × 7 × 7… 25 times. So remainder = 1 × 1 × 1…. 25 times = 1^{25}).
Example 2: What is the remainder, if 3^{63} is divided by 14.
Solution: If 3^{3} is divided by 14, the remainder is  1. So 3^{63} can be written as (3^{3})^{21}.
So the remainder is ( 1)^{21} =  1.
If the divisor is 14, the remainder  1 means 13. (14  1 = 13) by pattern method.
The binomial expansion of any expression of the form
(a + b)^{n} = ^{n}C_{o} a^{n} + ^{n}C_{1} a^{n1} × b^{1} + ^{n}C_{2} × a^{n2} × b^{2} ..... + ^{n}C_{n1} × a^{1} × b^{n1} + ^{n}C_{n} × b^{n}
Where ^{n}C_{o}, ^{n}C_{1}, ^{n}C_{2}, .... are all called the binomial coefficients
In general, ^{n}C_{r} = n!/r!(n  r)!
There are some fundamental conclusions that are helpful if remembered:
(a) There are (n + 1) terms.
(b) The first term of the expansion has only a.
(c) The last term of the expansion has only b.
(d) All the other (n  1) terms contain both a and b.
(e) If (a + b)^{n} is divided by a, then the remainder will be b^{n} such that b^{n} < a.
Example : What is the remainder if 7^{25} is divided by 6?
Solution.
(7)^{25} can be written (6 + 1)^{25}. So, in the binomial expansion, all the first 25 terms will have 6 in it. The 26th term is (1)^{25}. Hence, the expansion can be written 6x + 1. 6x denotes the sum of all the first 25 terms.
Since each of them is divisible by 6, their sum is also divisible by 6, and therefore, can be written 6x, where x is any natural number. So, 6x + 1 when divided by 6 leaves the remainder 1.
(OR)
When 7 divided by 6, the remainder is 1. So when 7^{25} is divided by 6, the remainder will be 1^{25} = 1.
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Introduction: Number System Video  53:18 min 
EvenOdd Multiplication Number Theory Video  02:09 min 
Cyclicity: Number Theory Video  03:24 min 
1. How can we find the remainder of a product using the Derivative of Remainder Theorem? 
2. How can we find the remainder of powers with the help of the Remainder Theorem? 
3. How can we apply the Binomial Theorem to find remainders? 
4. What is Wilson's Theorem? 
5. What is Fermat's Theorem? 
Introduction: Number System Video  53:18 min 
EvenOdd Multiplication Number Theory Video  02:09 min 
Cyclicity: Number Theory Video  03:24 min 

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