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Problems on Numbers Impotent Formulas:
1. Natural Numbers :
Counting numbers 1, 2, 3, 4, 5, .. are called natural numbers.
2. whole Numbers :
All counting numbers together with zero form the set of whole numbers.
3. Some Important Formulas :
I. ( 1 + 2 + 3 + .....+ n) = n (n + 1 ) / 2
II. (1 2 + 22 + 32 + ..... + n2) = n ( n + 1 ) (2n + 1) / 6
III. (1 3 + 23 + 33 + ..... + n3) = n2 (n + 1)2 / 4
4. Even Numbers:
Counting numbers 1, 2, 3, 4, 5, .. are called natural numbers.
2. whole Numbers :
All counting numbers together with zero form the set of whole numbers.
3. Some Important Formulas :
I. ( 1 + 2 + 3 + .....+ n) = n (n + 1 ) / 2
II. (1 2 + 22 + 32 + ..... + n2) = n ( n + 1 ) (2n + 1) / 6
III. (1 3 + 23 + 33 + ..... + n3) = n2 (n + 1)2 / 4
4. Even Numbers:
A Number Divisible by 2 is called an even number, such as 2,4,6,8,10....etc
5. Odd Numbers:
5. Odd Numbers:
A Number not divisible by 2 is called an odd number, such as 1,3,5,7,9,11..... etc
6. Prime Number:
6. Prime Number:
A Number, other than 1 is called a prime number, if it is divisible by 1 and itself only.
7. Some Basic Formulas:
i. (a + b)(a - b) = (a2 - b2)
ii. (a + b)2 = (a2 + b2 + 2ab)
iii. (a - b)2 = (a2 + b2 - 2ab)
iv. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
v. (a3 + b3) = (a + b)(a2 - ab + b2)
vi. (a3 - b3) = (a - b)(a2 + ab + b2)
vii. (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)
viii. When a + b + c = 0, then a3 + b3 + c3 = 3abc.
7. Some Basic Formulas:
i. (a + b)(a - b) = (a2 - b2)
ii. (a + b)2 = (a2 + b2 + 2ab)
iii. (a - b)2 = (a2 + b2 - 2ab)
iv. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
v. (a3 + b3) = (a + b)(a2 - ab + b2)
vi. (a3 - b3) = (a - b)(a2 + ab + b2)
vii. (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)
viii. When a + b + c = 0, then a3 + b3 + c3 = 3abc.
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