1. NUMBER SYSTEM

2. Chapters covered Numbers and divisibility rule H.C.F and L.C.M Factors
Multiples
Remainders
Cyclicity
Factorials

3. Divisibilty

4. Basic concept
Types of numbers
Real numbers : These numbers can be represented on the number line. For ex: any no ranges from -∞ to +∞
Imaginary numbers: These numbers can not be represented on the number line. For ex: i = √-1
Complex numbers: these numbers include both real and imaginary numbers.

5. Irrational numbers. E.g √2/3, √2, √7
Real numbers
Rational numbers. E.g 1, 2, 2/3, 3/4
Integers. E.g -∞, -4, 0 10, +∞
Whole numbers. E.g 0, 1, 2, … +∞
Natural numbers. E.g 1, 2,….+∞
Fractions. E.g ½, 2/3,
Irrational numbers. E.g √2/3, √2, √7

6. Some other classifications
Even numbers: these numbers are divisible by 2. Example: 2,4 ,6,…
Odd numbers: these numbers are not divisible by 2. Example: 1,3,5,…
Prime numbers: these numbers does not have a divisor apart from 1 and itself. Example: 2,3,5,11,29 etc.

7. Composite numbers: the numbers except 1 which are not prime are called composite numbers. Example: 4,6,20,100, etc.
Co-prime numbers: those numbers which do not have any common factor except 1 are called co-prime no. example: 15 and 26

8. How to check whether a no. is prime or not
Find nearest perfect square and will divide the no. by all prime no. till that nearest square if anyone divides the no. then that no. is not prime: For ex: take 151
Now the nearest perfect square is (12)²
So divide 151 by 2, 3, 5, 7, 11
Since none of the prime no. completely divides 151 so that means 151 is a prime no.

9. Divisibility Tests
A number is said to be divisible by another no. if the remainder
is 0
A number is divisible by 2 when its unit’s digit is even or 0
A number is divisible by 3 when the sum of digits is divisible by 3
A number is divisible by 4 when the number formed by the last two digits are either 0 or divisible by 4
A number is divisible by 5 when its unit’s digit is 5 or 0
A number is divisible by 6 when it is divisible by 2 and 3 both.

10. A number is divisible by 8 when when the number formed by the last three digits are either 0 or divisible by 8
A number is divisible by 9 when the sum of digits is divisible by 9
A number is divisible by 10 when its unit digit is 0
A number is divisible by 11 when the difference between the sum of the digits in the odd place and even place is 0 or a multiple of 11.
A number is divisible by 12 when is divisible by 3 and 4 both

11. Practice Questions
Q. If abc4d is divisible by 4, what is the value of d?
Q. A number 344ab5 is divisible by both 9 and 25. Find the number. (Given a + b<8)
Q. A number 1568X35Y is divisible by 88. What are the values of X and Y
Q. What is the remainder when is divisible by 9
a) b) 3
c) d) 4

12. Q. which of the following no. is divisible by 99?
a) b) 37332
c) d) 23337
Q. For n, a positive integer greater than 1, n(n²-1) is always divisible by
a) b) 12
c) d) 48

13. Q. What is the remainder when 78X85Y868 is divisible by 8
a) b) 3
c) d) 4
Q. If the number 786P86Q is divisible by 8 and 9 both, then values of P and Q are:
a) 4, b) 8,6
c) 6, d) 6,8