Tricks to Solve Number System Questions (Important Facts and Formula) for Bank, SSC, Railway and Other Government Job related Competitive Exam. Number System questions are commonly asked in all the Competitive Exam. In this Article we will Discuss some Tricks to Solve Number System Questions Quickly.
I.Numeral : In Hindu Arabic system, we
use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to
represent any number.
A group of digits, denoting a number is called a numeral.
We represent a number, say 689745132
as shown below :
6
|
8
|
9
|
7
|
4
|
5
|
1
|
3
|
2
|
We read it as :
'Sixty-eight crores, ninety-seven lacs, forty-five thousand, one hundred and
thirty-two'.
II Place Value or Local Value of a Digit in a
Numeral :
In the above numeral :
Place value of 2 is (2 x 1)
= 2; Place value of 3 is (3 x 10) = 30;
Place value of 1 is (1 x
100) = 100 and so on.
Place value of 6 is 6 x 108
= 600000000
III. Face Value : The face
value of a digit in a numeral is the value of the digit itself at whatever place it may be. In the
above numeral, the face value of 2 is 2; the face value of 3 is 3 and so on.
IV.TYPES OF
NUMBERS
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. Thus,
numbers. Thus,
(i) 0 is the only whole number
which is not a natural number.
(ii) Every natural number is a
whole number.
3.Integers : All natural numbers, 0
and negatives of counting numbers i.e.,
{…, -3,-2,-1, 0, 1, 2, 3,…..} together form the set of integers.
(i) Positive Integers : {1, 2, 3, 4, …..} is the
set of all positive integers.
(ii) Negative Integers : {- 1, - 2, - 3,…..} is the set of all negative
integers.
(iii) Non-Positive and
Non-Negative Integers : 0 is neither positive nor
negative. So, {0, 1, 2, 3,….} represents
the set of non-negative integers, while
{0, -1,-2,-3,…..} represents the set of non-positive
integers.
4. Even Numbers : A number divisible by 2 is called an even number, e.g., 2, 4, 6, 8,
10, etc.
5. Odd Numbers : A number not divisible by 2 is called an odd number. e.g., 1, 3, 5,
7, 9, 11, etc.
6. Prime Numbers : A number greater than 1 is called a prime number, if it has exactly
two factors, namely 1 and the number itself.
Prime numbers up to 100 are : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43,
47, 53, 59, 61, 67, 71, 73,
79, 83, 89, 97.
Prime numbers Greater than 100 : Let p be a given number greater
than 100. To find out whether it is prime or not, we use the following method :
Find a whole number nearly greater than the square root of p. Let k
> *jp. Test whether p is divisible by any prime number less than k. If yes,
then p is not prime. Otherwise, p is prime.
e.g,We have to find whether 191 is a prime number or not. Now, 14
> V191.
Prime numbers less than 14 are 2, 3, 5, 7, 11, 13.
191 is not divisible by any of them. So, 191 is a prime number.
7.Composite Numbers : Numbers greater
than 1 which are not prime, are known as composite numbers, e.g., 4, 6, 8, 9,
10, 12.
Note : (i) 1 is neither prime nor composite.
(ii) 2 is the only even number which is prime.
(iii) There are 25 prime numbers between 1 and 100.
8. Co-primes : Two numbers a and b are said to be co-primes, if their H.C.F. is 1.
e.g., (2, 3), (4, 5), (7, 9), (8, 11), etc. are co-primes,
V.TESTS OF
DIVISIBILITY
1. Divisibility By 2 : A number is divisible by 2, if its unit's digit is any of 0, 2, 4,
6, 8.
Ex. 84932 is divisible by 2, while 65935 is not.
2. Divisibility By 3 : A number is divisible by 3, if the sum of its digits is divisible
by 3.
Ex.592482 is divisible by 3, since sum
of its digits = (5 + 9 + 2 + 4 + 8 + 2) = 30, which is divisible by 3.
But, 864329 is not divisible by 3, since sum of its digits =(8 + 6 +
4 + 3 + 2 + 9) = 32, which is not divisible by 3.
3. Divisibility By 4 : A number is divisible by 4, if the number formed by the last two
digits is divisible by 4.
Ex. 892648 is divisible by 4, since the number formed by the last
two digits is
48, which is divisible by 4.
But, 749282 is not divisible by 4, since the number formed by the
last tv/o digits is 82, which is not divisible by 4.
4. Divisibility By 5 : A number is divisible by 5, if its unit's digit is either 0 or 5.
Thus, 20820 and 50345 are divisible by 5, while 30934 and 40946 are not.
5. Divisibility By 6 : A number is divisible by 6, if it is divisible by both 2 and 3. Ex.
The number 35256 is clearly divisible by 2.
Sum of its digits = (3 + 5 + 2 + 5 + 6) = 21, which is divisible by
3. Thus, 35256 is divisible by 2 as well as 3. Hence, 35256 is divisible by 6.
6. Divisibility By 8 : A number is divisible by 8, if the number formed by the last
three digits of the given number is divisible by 8.
Ex. 953360 is divisible by 8, since the number formed by last three
digits is 360, which is divisible by 8.
But, 529418 is not divisible by 8, since the number formed by last
three digits is 418, which is not divisible by 8.
7. Divisibility By 9 : A number is divisible by 9, if the sum of its digits is divisible
by 9.
Ex. 60732 is divisible by 9, since sum of digits * (6 + 0 + 7 + 3 +
2) = 18, which is divisible by 9.
But, 68956 is not divisible by 9, since sum of digits = (6 + 8 + 9 +
5 + 6) = 34, which is not divisible by 9.
8. Divisibility By 10 : A number is divisible by 10, if it ends with 0.
Ex. 96410, 10480 are divisible by 10, while 96375 is not.
9. Divisibility By 11 : 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 either 0 or a
number divisible by 11.
Ex. The number 4832718 is divisible by
11, since :
(sum of digits at odd places) - (sum of digits at even places)
- (8 + 7 + 3 + 4) - (1 + 2 + 8) = 11, which is divisible by 11.
10.
Divisibility By 12 ; A number is divisible by 12, if it is divisible by
both 4 and
3.
Ex. Consider the number 34632.
(i) The number formed by last two digits is 32, which is divisible
by 4,
(ii) Sum of digits = (3 + 4 + 6 + 3 + 2) = 18, which is divisible by
3. Thus, 34632 is divisible by 4 as well as 3. Hence, 34632 is divisible by 12.
11.
Divisibility By 14 : A number is divisible by 14, if it is divisible by
2 as well as 7.
12.
Divisibility By 15 : A number is divisible by 15, if it is divisible by
both 3 and 5.
13.
Divisibility By 16 : A number is divisible by 16, if the number formed
by the last4 digits is divisible by 16.
Ex.7957536 is divisible by 16, since the
number formed by the last four digits is 7536,
which is divisible by 16.
14. Divisibility By 24 : A given number is divisible by 24, if it is divisible by both3 and 8.
15. Divisibility By 40 : A given number is divisible by 40, if it is divisible by both
5 and 8.
16. Divisibility By 80 : A given number is divisible by 80, if it is divisible by both 5 and
16.
Note : If a number is divisible by p as
well as q, where p and q are co-primes, then the given number is divisible by
pq.
If p arid q are not co-primes, then the given number need not be
divisible by pq,
even when it is divisible by both p and q.
Ex. 36 is divisible by both 4 and 6, but
it is not divisible by (4x6) = 24, since
4 and 6 are not co-primes.
VI. MULTIPLICATION BY SHORT
CUT METHODS
1. Multiplication By Distributive
Law :
(i) a x (b + c) = a x b + a x c
(ii) ax(b-c) = a x b-a x c.
Ex.
(i) 567958 x 99999 = 567958 x (100000 - 1)
= 567958 x 100000 - 567958 x 1 = (56795800000 - 567958) =
56795232042. (ii) 978 x 184 + 978 x 816 = 978 x (184 + 816) = 978 x 1000 =
978000.
2. Multiplication of a Number
By 5n : Put n zeros to the right of the multiplicand and divide the number
so formed by 2n
Ex. 975436 x 625 = 975436 x 54= 9754360000 = 609647600
VII. BASIC FORMULA
1. (a + b)2 = a2 + b2 + 2ab 2. (a - b)2 =
a2 + b2 - 2ab
3. (a + b)2 - (a - b)2 = 4ab 4. (a + b)2 +
(a - b)2 = 2 (a2 + b2)
5. (a2 - b2)
= (a + b) (a - b)
6. (a + b + c)2 =
a2 + b2 + c2 + 2 (ab + bc + ca)
7. (a3 + b3)
= (a +b) (a2 - ab + b2) 8. (a3 - b3) = (a
- b) (a2 + ab + b2)
9. (a3 + b3 + c3 -3abc) = (a + b +
c) (a2 + b2 + c2 - ab - bc - ca)
10. If a + b + c = 0, then a3 + b3 + c3
= 3abc.
VIII. DIVISION ALGORITHM OR
EUCLIDEAN ALGORITHM
If we divide a given number by another number, then :
Dividend = (Divisor x Quotient) + Remainder
IX. DIVISIBILITY SHORT
TRICKS
{i) (xn - an ) is divisible by (x - a) for all
values of n.
(ii) (xn
- an) is divisible by (x + a) for all even values of n.
(iii) (xn + an)
is divisible by (x + a) for all odd values of n.
X. PROGRESSION
A succession of numbers formed and arranged in a definite order
according to certain definite rule, is called a progression.
1. Arithmetic Progression (A.P.) : If
each term of a progression differs from its preceding term by a constant, then
such a progression is called an arithmetical progression. This constant
difference is called the common difference of the A.P.
An A.P. with first term a and common difference d is given by a, (a
+ d), (a + 2d),(a + 3d),.....
The nth term of this A.P. is given by Tn =a (n - 1) d.
The sum of n terms of this A.P.
Sn = n/2 [2a + (n - 1) d] = n/2 (first term + last term).
SOME IMPORTANT RESULTS :
(i) (1 + 2 + 3 +…. + n)
=n(n+1)/2
(ii) (l2 + 22 + 32 + ... + n2)
= n (n+1)(2n+1)/6
(iii) (13 + 23
+ 33 + ... + n3) =n2(n+1)2
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