Properties of number:
1. Commutative Property: This property tells us we can add and multiply numbers in any order.
(i) a + b + c = b + a + c
Ex. 3 + 5 + 6 = 5 + 3 + 6
In left hand side we add 3 & 5, and then we will get 8
In right hand side we add 5 & 3, and then we will also get 8.
8 + 6 = 6 + 8
14 = 14
L.H.S and R.H.S both are equal.
(ii) axbxc = bxcxa
5 x 6 x 7 = 6 x 7 x 5
In left hand side we multiply 5 & 6, and then we will get 30
In right hand side we add 6 & 7, and then we will also get 42.
30 x 7 = 42 x 5
210 = 210.
LHS and RHS both are equal.
2. Associative Property: Both addition and multiplication can be done with two numbers at a time.
(i) a +(b + c)= (a + b) +c
Ex. 5 + (6 + 8) = (5 + 6) + 8
5 + 14 = 11 + 8
19 = 19
Hence proved, LHS and RHS both are equal.
(ii) ax (b x c) = (a x b) x c
Ex. 5 x (6 x 8) = (5 x 6) x 8
5 x 48 = 30 x 8
240 = 240
Hence proved, L.H.S and R.H.S both are equal.
3. Distributive Property: In this property "multiplication distributes over addition".
a x (b +c) = a x b + a x c
Ex. 5 x (6 + 8) = 5 x 6 + 5 x 8
5 x 14 = 30 + 40
70 = 70
Hence proved, L.H.S and R.H.S both are equal.
4. Identity property: It tells us that zero added to any number is the number itself. Zero is called the "additive identity." The identity property for multiplication tells us that the number 1 multiply by any number gives the number itself. The number 1 is called the "multiplicative identity."
Ex. a +0 = a
a.1 = a
5. Additive Inverse: The additive inverse of a number ‘a’ is the number that when added to ‘a’, yields zero. This number is also known as the opposite (number), sign change, and negation.
Ex. a + (-a) = 0.
6. Multiplicative Inverse: It is reciprocal of a, denoted 1/a or a-1.
Ex. a.1/a = 1
7. Additive Inverse: In this property, reverses the sign only.
Ex. a+ (-a) = 0.
Divisibility Rules: Divisibility means that you are able to divide a number evenly without using the long division.
Divisibility by 2: When the unit place digit of any number is 0,2,4,6 or 8 then the number is divisible by 2.
Ex. 10, 12, 1324, 13564, 434432, 9429372, etc.
Divisibility by 3: When the sum of digits of a number is divisible by 3 then the number is also divisible by 3.
Ex. 465981: 4+6+5+9+8+1= 33, which is divisible by 3, so 465981 must be divisible by 3.
Divisibility by 4: When the number formed by the last two digits of given number is divisible by 4 or it must be two or more zeros then the number is divisible by 4.
Ex.448, 1428, 33700, 4387920, etc.
Divisibility by 5: When the last digit of any number is either ‘5’ or ‘0’ is divided by 5.
Ex. 6742735, 3749370, 27629345, etc.
Divisibility by 6: When the number is divisible by 2 and 3 both then the number must be divisible by 6.
Ex. 356478, 82458474, 73834626, are divisible by 6.
Divisibility by 7: When the difference between the number formed by the digits other than the units digit and twice the units digit is either 0 or multiple of 7.
Ex. 798 is divisible by 7 because 79-2x8=79-16=63 is divisible by 7 and 798 is also divisible by 7.
Divisibility by 8: When the number formed by the last three digits of given number is divisible by 8 or it must be three or more zeros then the number is divisible by 8.
Ex. 848, 10128,323000, 438720, etc.
Divisibility by 9: When the sum of digits of a number is divisible by 9 then the number is also divisible by 9.
Ex. 39469581: 3+9+4+6+9+5+8+1= 45, which is divisible by 9, so 39469581 must be divisible by 9.
Divisibility by 11: When the sum of digits at odd and even places is equal or differs by a number divisible by 11, then the number is divisible by 11.
Ex. 7004844 is divisible by 11, since
(Sum of digits at odd places) - (Sum of places at even places) = (7+0+8+4) - (0+4+4) = 19 - 8= 11.
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