# Shortcuts for Number Theory

### NUMBER THEORY

1. Sum of natural numbers from 1 to n

e.g Sum of natural numbers from 1 to 40 = 40(40+1)/2 = 820

2. Sum of squares of first n natural numbers is =

3. Sum of the squares of first n even natural numbers is

4. Sum of cubes of first n natural numbers is

5. Any number N can be represented in the decimal system of number as

#### Important Formulas

i. ( a +  b )( a –  b ) = ( a 2 –  b 2 )

ii. ( a +  b ) 2 = ( a 2 + b 2+ 2 ab )

iii. ( a –  b )2 = ( a2 + b 2 – 2 ab )

iv. ( a +  b +  c ) 2 =  a2 + b2 +  c 2 + 2 ( ab +  bc +  ca )

v. ( a 3 +  b3 ) = ( a +  b )( a 2 –  ab +  b 2 )

vi. ( a 3 –  b 3 ) = ( a –  b )( a 2 +  ab +  b 2 )

vii. ( a 3 +  b 3 +  c 3 – 3 abc ) = ( a +  b +  c )( a 2 +  b 2 +  c 2 –  ab –  bc –  ac )

viii. When  a +  b +  c = 0, then  a 3 +  b 3 +  c 3 = 3 abc .

xi. ( a + b ) 2 = ( a 2 + b 2 + 2 ab ) =(a – b)2 + 4ab

x. ( a – b ) 2 = ( a 2 + b 2 – 2 ab ) = (a + b)2 – 4ab

#### Some more tips:

1) k(a + b + c) = ka + kb + kc

2) (a + b) (c + d) = ac + ad + bc + bd

3) (x + a) (x + b) = x2 + (a + b)x + ab

4) ( a + b ) 2 – ( a – b ) 2 = 4ab

5) ( a + b ) 2 – ( a – b ) 2 = 2(a2 + b2)

6) (a + b)3 = a3 + b3 + 3ab(a + b) = a3 + 3a2b +3ab2 + b3

7) (a – b)3 = a3 – b3 – 3ab(a – b) = a3 – 3a2b +3ab2 – b3

8) 1/a + 1/b = (a + b)/ ab

9) (x + a)(x + b)(x + c) = x3 + (a +b +c)x2 + (ab + bc + ca)x +abc

10) (a + b + c)3 = a3 + b3 + c3 + 3a2b + 3a2c + 3b2a +3b2c + 3c2a + 3c2b + 6abc