Simplification:
Read the following points carefully.
TYPES OF NUMERALS:
1). Natural Numbers: These are counting numbers. For Ex. 1,2,3,4,5……
a) Natural numbers are denoted by ‘N’.
b) All natural numbers are positive only.
c) 0 is not a natural number. The smallest natural number is ‘1’
2). Whole Numbers: Whole numbers having ‘0’ and natural numbers. Ex.0, 1, 2, 3, 4…….
a) Denoted by ‘W’
3) Integers: Negative numbers and whole numbers forms integers. Ex. -3, -2, -1, 0, 1, 2, 3….
a) Denoted by ‘I’
b) ‘0’ number is neither positive nor negative.
4) Even and Odd Numbers: The counting numbers which are divisible by 2 then those numbers called Even numbers otherwise Odd numbers.
a) Ex. For even numbers: 2, 4, 6, 8,10 ….
b) Ex. For odd numbers: 1, 3, 5, 7,9 ….
5) Prime Numbers: These numbers having only two factors ‘1’ and itself.
Ex. 2, 3, 5, 7, 11, 13 …. 2 is the even number which is prime and prime number is always greater than 1.
6) Rational and Irrational Numbers: Rational numbers are represented in the form of (a/b) where b is not equal to ‘0’, a and b are integers. Ex. (1/2), (2/5), (3/8)…
Irrational numbers cannot be represent in the form of (a/b). Ex. Square roof of 2
7) Composite Numbers: These numbers are not prime numbers and having at least one factor other than ‘1’ and itself.For Ex. 4, 8, 12…
Divisibility Rules:
1. Divisibility by 2: If the number is having either ‘0’ or even number as a last digit then that number is divisible by 2.
Example: 24, 40, 68, 122
2. Divisibility by 3: When sum of the digits of a number is divisible by 3, then the number is divisible by 3.
Example: 159 1 + 5 + 9 = 15 15 is divisible by 3.
3. Divisibility by 4: If the last two digits of a number is divisible by 4, then that number is divisible by 4.
Example: 124 24 is divisible by 4
If the number is having two or more zeroes at the end then also its divisible by 4.
Example: 1200 it’s having two zeroes at the end so its divisible by 4.
4. Divisibility by 5: If the number is having ‘0’ or ‘5’ at the end then it’s divisible by 5.
Example: 150, 205, 300
5. Divisibility by 6: If the number is divisible by 2 and 3then that number should divisible 6.
Example: 36, 54, 60
6. Divisibility by 7: when the difference between twice the digit at last place and the number formed by other digits is either ‘0’ or multiple of ‘7’.
Example: 147 14-(2*7) = 0 144 is divisible by 7
7. Divisibility by 8: If the number made by last three digits is divisible by 8, the number is divisible by 8.
Example: 24032 032 is divisible 8
8. Divisibility by 9: It’s similar to divisible by 3 rule. When sum of the all the digits of a number is divisible by 9, then the number is divisible by 9.
Example: 4374 4 + 3 + 7 + 4 = 18 18 is divisible by 9
9. Divisibility by 10: If the number is end with ‘0’ then it’s divisible by 10.
Example: 50, 110, 2000
10. Divisibility by 11: If the sum of digits at odd and even places are equal or differ by a number which is divisible by 11 then the number is divisible by 11.
Example: 216282 sum at even places(2+6+8) = 16; sum at odd places (1+2+2) = 5
Difference: (16-5) = 11. So 216282 is divisible by 11.
Read the following points carefully.
TYPES OF NUMERALS:
1). Natural Numbers: These are counting numbers. For Ex. 1,2,3,4,5……
a) Natural numbers are denoted by ‘N’.
b) All natural numbers are positive only.
c) 0 is not a natural number. The smallest natural number is ‘1’
2). Whole Numbers: Whole numbers having ‘0’ and natural numbers. Ex.0, 1, 2, 3, 4…….
a) Denoted by ‘W’
3) Integers: Negative numbers and whole numbers forms integers. Ex. -3, -2, -1, 0, 1, 2, 3….
a) Denoted by ‘I’
b) ‘0’ number is neither positive nor negative.
4) Even and Odd Numbers: The counting numbers which are divisible by 2 then those numbers called Even numbers otherwise Odd numbers.
a) Ex. For even numbers: 2, 4, 6, 8,10 ….
b) Ex. For odd numbers: 1, 3, 5, 7,9 ….
5) Prime Numbers: These numbers having only two factors ‘1’ and itself.
Ex. 2, 3, 5, 7, 11, 13 …. 2 is the even number which is prime and prime number is always greater than 1.
6) Rational and Irrational Numbers: Rational numbers are represented in the form of (a/b) where b is not equal to ‘0’, a and b are integers. Ex. (1/2), (2/5), (3/8)…
Irrational numbers cannot be represent in the form of (a/b). Ex. Square roof of 2
7) Composite Numbers: These numbers are not prime numbers and having at least one factor other than ‘1’ and itself.For Ex. 4, 8, 12…
Divisibility Rules:
1. Divisibility by 2: If the number is having either ‘0’ or even number as a last digit then that number is divisible by 2.
Example: 24, 40, 68, 122
2. Divisibility by 3: When sum of the digits of a number is divisible by 3, then the number is divisible by 3.
Example: 159 1 + 5 + 9 = 15 15 is divisible by 3.
3. Divisibility by 4: If the last two digits of a number is divisible by 4, then that number is divisible by 4.
Example: 124 24 is divisible by 4
If the number is having two or more zeroes at the end then also its divisible by 4.
Example: 1200 it’s having two zeroes at the end so its divisible by 4.
4. Divisibility by 5: If the number is having ‘0’ or ‘5’ at the end then it’s divisible by 5.
Example: 150, 205, 300
5. Divisibility by 6: If the number is divisible by 2 and 3then that number should divisible 6.
Example: 36, 54, 60
6. Divisibility by 7: when the difference between twice the digit at last place and the number formed by other digits is either ‘0’ or multiple of ‘7’.
Example: 147 14-(2*7) = 0 144 is divisible by 7
7. Divisibility by 8: If the number made by last three digits is divisible by 8, the number is divisible by 8.
Example: 24032 032 is divisible 8
8. Divisibility by 9: It’s similar to divisible by 3 rule. When sum of the all the digits of a number is divisible by 9, then the number is divisible by 9.
Example: 4374 4 + 3 + 7 + 4 = 18 18 is divisible by 9
9. Divisibility by 10: If the number is end with ‘0’ then it’s divisible by 10.
Example: 50, 110, 2000
10. Divisibility by 11: If the sum of digits at odd and even places are equal or differ by a number which is divisible by 11 then the number is divisible by 11.
Example: 216282 sum at even places(2+6+8) = 16; sum at odd places (1+2+2) = 5
Difference: (16-5) = 11. So 216282 is divisible by 11.
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