NUMBER SYSTEMS
Concepts and formulas:

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 = ( a 2 + b 2 - 2 ab )

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

v.   ( a 3 +  b 3 ) = ( 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

Tests of Divisibility :

##### 9. A number is divisible by 11 if the difference of the sum of its digits at odd places and                         the sum of its digits at even places, is divisible by 11.

Some more tips:
1) A number is divisible by 12, when it is divisible by both 3 and 4.

2) A number is divisible by 25, when the last two digits are 00 or divisible by 25

3) A number is divisible by 125, if the last three digits are 000 or divisible by 125

4) A number is divisible by 27, if the sum of the digits of the number is divisible by 27.

5) A number is divisible by 125, if the number formed by last three digits is divisible by 125.

6) Number of the form 10(n-1) (where 'n' is a natural number) is always divisible by 9
if 'n' is even, such numbers are divisible by 11 also.

H.C.F and L.C.M :
H.C.F stands for Highest Common Factor. The other names for H.C.F are Greatest Common Divisor (G.C.D) and Greatest Common Measure (G.C.M).  The H.C.F. of two or more numbers is the greatest number that divides each one of them exactly.  Two numbers are said to be co-prime if their H.C.F. is 1.  The least number which is exactly divisible by each one of the given numbers is called their L.C.M.
Finding L.C.M and H.C.F of Fractions
LCM= LCM of the numerators
HCF of the denominators
HCF= HCF of the numerators
LCM of the denominators
Product of two numbers = Product of their H.C.F. and L.C.M.
9 and 10 used
Examples:
1. 117 * 117 + 83 * 83 = ?
a) 20698            b) 20578        c) 21698        d) 21268
1. (1/4)3 + (3/4)3 + 3(1/4)(3/4)(1/4 + 3/4) =?
a) 1/64              b)27/64     c) 49/64         d)1
1. When 26854 and 27584 are divided by a certain two digit prime number, the remainder obtained is 47. Which of the following choices is a possible value of the divisor?
a) 61         b) 71         c) 73         d) 89
1. The 7th digit of (202)3 is
a) 2         b) 4         c) 8         d) 6
1. H.C.F. of two numbers is 16. Which one of the following can never be their L.C.M
a) 32         b) 80         c) 64         d) 60
1. What is the remainder when 9 + 92 + 93 + .... + 98 is divided by 6?
a) 3          b) 2         c) 0         d)5
1. The sum of the first 100 natural numbers is divisible by
a) 2, 4 and 8      b) 2 and 4                c)2 only         d)none of these
1. For what value of 'n' will the remainder of 351n and 352n be the same when divided by 7?
a) 2         b)3          c)6          d)4
1. Let n be the number of different 5 digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is the value of n?
a) 144         b) 168           c)192                 d)none of these
1. Find the greatest number of five digits, which is exactly divisible by  7, 10, 15, 21 and 28.
a) 99840             b) 99900          c)99960      d) 99990
Answer Key:
1.B;  2.D;  3.C;  4.C;  5.D;  6.C;  7.C;  8.B;  9.C;  10.C