Top Ad unit 728 × 90

Recent

recent

NUMBER SYSTEM -- A FEW PROBLEMS


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 : 
 
1. A number is divisible by 2 if it is an even number. 
2. A number is divisible by 3 if the sum of the digits is divisible by 3
3. A number is divisible by 4 if the number formed by the last two digits is divisible by 4. 
4. A number is divisible by 5 if the unit digit is either 5 or 0. 
5. A number is divisible by 6 if the number is divisible by both 2 and 3. 
6. A number is divisible by 8 if the number formed by  last three digits is divisible by 8. 
7. A number is divisible by 9 if the sum of the digits is divisible by 9. 
8. A number is divisible by 10 if the unit digit is 0. 
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
NUMBER SYSTEM -- A FEW PROBLEMS Reviewed by sambasivan srinivasan on 6:34:00 AM Rating: 5

No comments:

All Rights Reserved by Bank Exams © 2009
Technology PartnerNiralcube

Contact Form

Name

Email *

Message *

Powered by Blogger.