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Number System
3
CHAPTER 1
Number System
A Civil Servant should be well-versed in basics of Number System. In the Civil Services Aptitude Test Paper
2, in Basic Numeracy, certainly there will be asked some questions based on types of, and operations on numbers.
In Indian system, numbers are expressed by means of symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, called digits. Here, 0
is called insignificant digit whereas 1, 2, 3, 4, 5, 6, 7, 8, 9 are called significant digits. We can express a number in
two ways.
Notation: Representing a number in figures is known as notation as 350.
Numeration: Representing a number in words is known as numeration as ‘Five hundred and forty five’.
Place Value (Indian)
Crore
Ten Crore
100000000
108
Lakh
Crore
10000000
107
Ten Lakhs
1000000
106
Lakh
100000
105
Thousand
Ten Thousands
10000
104
Unit
Thousand
1000
103
Hundred
100
102
Tens
10
101
One
1
100
Place Value (International)
Million
Hundred
Millions
100000000
108
Thousand
Ten Millions
One Million
10000000
102
1000000
106
Hundred
Thousands
100000
105
Ten
Thousands
10000
104
Unit
Thousand
Hundred
Tens
One
1000
103
100
102
10
101
1
100
Face Value and Plac e Value of a Digit
Face Value: It is the value of the digit itself eg, in 3452, face value of 4 is ‘four’, face value of 2 is ‘two’.
Place Value: It is the face value of the digit multiplied by the place value at which it is situated eg, in 2586,
place value of 5 is 5 × 102 = 500.
Num be r C a te g o rie s
Natural Numbers (N): If N is the set of natural numbers, then we write N = {1, 2, 3, 4, 5, 6,…}
The smallest natural number is 1.
Whole Numbers (W): If W is the set of whole numbers, then we write W = {0, 1, 2, 3, 4, 5,…}
The smallest whole number is 0.
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Number System
Integers (I): If I is the set of integers, then we write I = {– 3, –2, –1, 0, 1, 2, 3, …}
Rational Numbers: Any number which can be expressed in the form of p/q, where p and q are both integers
and q # 0 are called rational numbers.
eg,
3  7
, ,5,  2
2 9
There exists infinite number of rational numbers between any two rational numbers.
Irrational Numbers Non-recurring and non-terminating decimals are called irrational numbers. These
p
numbers cannot be expressed in the form of q .
eg,
3, 5, 29,
Real Numbers: Real number includes both rational and irrational numbers.
B asic Ru les o n Na tura l Numbers
1. One digit numbers are from 1 to 9. There are 9 one digit numbers. ie, 9 × 100.
2. Two digit numbers are from 10 to 99. There, are 90 two digit numbers. ie, 9 × 10.
3. Three digit numbers are from 100 to 199. There are 900 three digit numbers ie, 9 × 102.
In general the number of n digit numbers are 9 × 10(n–1)
4. Sum of the first n, natural numbers ie, 1 + 2 + 3 + 4 + … + n =
n  n  1
2
5. Sum of the squares of the first n natural numbers ie. 12 + 23 + 32 + 42 + …+ n2 =
6. Sum of the cubes of the first n natural
ie,
12
+
23 +
32 +
…+
n3
n  n  1  2n  1 
6
= 

 n  n  1 2 

2



Example: What is the value of 51 + 52 + 53 + … + 100 ?
Solution.
51 + 52 + 33 + ... + 100 = (1 + 3 + …+ 100) – (1 + 2 + 3 + ... + 50)
=
100  101 50  51

= 5050 —1275 = 3775
2
2
D iffe ren t Type s o f Num be rs
Even Numbers: Numbers which are exactly divisible by 2 are called even numbers.
eg,
– 4, – 2, 0, 2, 4…
Sum of first n even numbers = n (n + 1)
Odd Numbers: Numbers which are not exactly divisible by 2 are called odd numbers.
eg,
– 5, –3, –1, 0, 1, 3, 5…
Sum of first n odd numbers = n2
Prime Numbers: Numbers which are divisible by one and itself only are called prime numbers.
eg,
2, 3, 5, 7, 11…
• 2 is the only even prime number.
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Number System
• 1 is not a prime number because it has two equal factors.
• Every prime number greater than 3 can be written in the form of (6K + 1) or (6K – 1) where K is an integer.
® There are 15 prime numbers between 1 and 50 and l0 prime numbers between 50 and 100.
Relative Prime Numbers: Two numbers are said to be relatively prime if they do not have any common
factor other than 1.
eg,
(3, 5), (4, 7), (11, 15), (15, 4)…
Twin Primes: Two prime numbers which differ by 2 are called twin primes.
eg,
(3, 5), (5, 7), (11, 13),…
Composite Numbers Numbers which are not prime arc called composite numbers
eg,
4, 6, 9, 15,…
1 is neither prime nor composite.
Perfect Number: A number is said to be a perfect number, if the sum of all its factors excluding itself is
equal to the number itself. eg, Factors of 6 are 1, 2, 3 and 6.
Sum of factors excluding 6 = 1 + 2 + 3 = 6.
 6 is a perfect number.
Other examples of perfect numbers are 28, 496, 8128 etc.
Rules for Divisibility
Divisibility by 2: A number is divisible by 2 when the digit at ones place is 0, 2, 4, 6 or 8.
eg,
3582, 460, 28, 352, ....
Divisibility by 3: A number is divisible by 3 when sum of all digits of a number is a multiple of 3.
eg,
453 = 4 + 5 + 3 = 12.
12 is divisible by 3 so, 453 is also divisible by 3.
Divisibility by 4: A number is divisible by 4, if the number formed with its last two digits is divisible by 4. eg,
if we take the number 45024, the last two digits form 24. Since, the number 24 is divisible by 4, the number 45024
is also divisible by 4.
Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
eg,
10, 25, 60
Divisibility by 6: A number is divisible by 6, if it is divisible both by 2 and 3.
eg,
48, 24, 108
Divisibility by 7: A number is divisible by 7 when the difference between twice the digit at ones place and
the number formed by other digits is either zero or a multiple of 7.
eg,
658
65 – 2 × 8 = 65 – 16 = 49
As 49 is divisible by 7 the number 658 is also divisible by 7.
Divisibility by 8: A number is divisible by 8, if the number formed by the last 3 digits of the number is
divisible by 8. eg, if we take the number 57832, the last three digits form 832. Since, the number 832 is divisible
by 8, the number 57832 is also divisible by 8..
Divisibility by 9: A number is divisible by 9, if the sum of all the digits of a number is a multiple of 9.
eg,
684 = 6 + 8 + 4 = 18.
18 is divisible by 9 so, 684 is also divisible by 9.
Divisibility by 10: A number is divisible by 10, if its last digit is 0. eg, 20, 180, 350,….
Divisibility by 11: When the difference between the sum of its digits in odd places and in even places is
either 0 or a multiple of 11.
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Number System
eg,
30426
3 + 4 + 6 = 13
0+2 =2
13 – 2 = 11
As the difference is a multiple of 11 the number 30426 is also divisible by 11.
‘Smart’ Facts
• If p and q are co-primes and both are factors of a number K, then their product p x q will also be a factor of r. eg,
Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 prime factors of 24 are 2 and 3, which are co-prime also. Product of
2 × 3 = 6, 6 is also a factor of 24.
• If ‘p’ divides ‘q’ and ‘r’, then p’ also divides their sum or difference. eg, 4 divides 12 and 20. Sum of 12 and 20 is 32
which is divisible by 4. Difference of 20 and 12 is 8 which is divisible by 4.
• If a number is divisible by another number, then it must be divisible by each of the factors of that number. 48 is
divisible by 12. Factors of 12 are 1, 2, 3, 4, 6, 12. So, 48 is divisible by 2, 3, 4 and 6 also.
Divis ion on Nu mbers
In a sum of division, we have four quantities.
They are (i) Dividend, (ii) Divisor, (iii) Quotient and (iv) Remainder. These quantities are connected by a
relation.
(a) Dividend = Divisor × Quotient + Remainder.
(b) Divisor = (Dividend – Remainder) ÷ Quotient.
(c) Quotient = (Dividend – Remainder) – Divisor.
Example 2: In a sum of division, the quotient is 110, the remainder is 250, the divisor is equal to the sum of
the quotient and remainder. What is the dividend ?
Solution.
Divisor = (110 + 250) = 360
Dividend = (360 × 110) + 250 = 39850
Hence, the dividend is 39850.
Example 3: Find the number of numbers upto 600 which are divisible by 14.
Solution. Divide 600 by 13, the quotient obtained is 46. Thus, there are 46 numbers less than 600 which are
divisible by 14.
F ac t o rs a nd Mult iple s
Factor: A number which divides a given number exactly is called a factor of the given number,
eg,
24 = 1 × 24, 2 × 12, 3 × 8, 4 × 6
Thus, 1, 2, 3, 4, 6, 8, 12 and 24 are factors of 24.
• 1 is a factor of every number
• A number is a factor of itself
• The smallest factor of a given number is 1 and the greatest factor is the number itself.
• If a number is divided by any of its factors, the remainder is always zero.
• Every factor of a number is either less than or at the most equal to the given number.
• Number of factors of a number are finite.
Number of Factors of a Number: If N is a composite number such that N = am bn c°... where a, b, c ... are
prime factors of N and m, n, o ... are positive integers, then the number of factors of N is given by the expression
(m + 1) (n + 1) (o + 1)
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Number System
Example 4: Find the number of factors that 224 has.
Solution. 224 = 25 × 71
Hence, 224 has (5 + 1) (1 + 1) = 6 × 2 = 12 factors.
Multiple: A multiple of a number is a number obtained by multiplying it by a natural number eg,
Multiples of 5 are 5, 10, 15, 20
Multiples of 12 are 12, 24, 36, 48
•
•
•
•
Every number is a multiple of 1.
The smallest multiple of a number is the number itself.
We cannot find the greatest multiple of a number.
Number of multiples of a number are infinite.
EXERCISE
1. Evaluate:
2.
3.
4.
5.
6.
7.
9 3  5  5 4 ÷10
 3  5  2 × 4 ÷ 2
(a) 9/10
(b) –8/17
(c) –16/19
(d) 4/7
The sum of three consecutive natural numbers
each divisible by 3 is 72. What is the largest
among them?
(a) 25
(b) 26
(c) 27
(d) 30
55% of a number is more than one-third of that
number by 52. What is two-fifth of that number?
(a) 96
(b) 240
(c) 144
(d) 142
The digits of a two-digit number are in the ratio
of 2 : 3 and the number obtained by interchanging
the digits is bigger than the original number by
27. What is the original number?
(a) 63
(b) 48
(c) 96
(d) 69
What least number would be subtracted from
427398 so that the remaining number is divisible
by 15?
(a) 13
(b) 3
(c) 16
(d) 11
If 45% of a number is added to the another
number, the first number becomes 135 times of
the another number. What is the ratio of these
two numbers?
(a) 8 : 7
(b) 3 : 2
(c) 7 : 8
(d) None of these
A number gets reduced to its one-third when 48
is subtracted from it. What is two -third of that
number?
8.
9.
10.
11.
12.
(a) 22
(b) 76
(c) 36
(d) 48
One-fifth of a number is equal to 5/8th of another
number. If 35 is added to the first number, it
becomes four times of the second number. Find
the second number.
(a) 39
(b) 70
(c) 40
(d) 25
Five- eighth of three-tenth of four-ninth of a
number is 45. What is the number?
(a) 470
(b) 550
(c) 560
(d) 540
Which of the following numbers should be added
to 11158 to make it exactly divisible by 77?
(a) 9
(b) 8
(c) 6
(d) 7
n
If n is odd, (11) + 1 is divisible by:
(a) 11 + 1
(b) 11 – 1
(c) 11
(d) 10 + 1
X and Y are positive integers and X is less than
Y. If X2 + Y2 equals two times 5 and XY equals
two times 2, X / Y will equal:
(a) 3
(b) 1 2
(c) 1
(d) 2
13. Find unit digit in (515)31 + (515)90:
(a) 0
(b) 5
(c) 1
(d) 4
 n  n  1 

14. Consider the number, 
where n is a
2 

positive integer. Which of the following is
necessarily false?
2
 n  n  1 
 is divisible by the sum of first n
(a) 
2 

natural numbers.
2
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Number System
 n  n  1 
 is divisible by the sum of the cubes
(b) 
2 

of first n natural numbers.
(c) {n(n + 1)}2 is always even.
2
 n  n  1 
 is never divisible by 237.
(d) 
2 

The number 899 is:
(a) a number with 5 factors
(b) a number with 4 factors
(c) a number with more than 4 factors
(d) a perfect cube
The numerator of a fraction is multiple of two
numbers. One of the numbers is greater than the
other by 2. The greater number is smaller than
the denominator by 1. If the denominator is given
as 5 + c(c is a constant), then the minimum value
of the fraction is:
(a) 2/3
(b) –2
(c) –1/2
(d) 1/2
Find the number which when multiplied by 13 is
increased by 180:
(a) 20
(b) 15
(c) 124
(d) 5
Find the number of divisors of 10800.
(a) 57
(b) 60
(c) 72
(d) 62
The sum of the digits in a two-digit number is 5.
If 9 is subtracted from the number, the result is
the number with the digits reversed. The number
is:
(a) 23
(b) 24
(c) 41
(d) 14
Three consecutive numbers such that twice the
first, 3 times the second and 4 times the third
together make 182. The numbers in question are:
(a) 18, 22 and 23
(b) 18, 19 and 20
(c) 19, 20 and 21
(d) 20, 21 and 22
Find the product of place value and face value of
5 in 65231.
(a) 28000
(b) 25000
(c) 27000
(d) 26000
Find the least value of K so that 39 K20 is divisible
by 3.
(a) 1
(b) 3
(c) 5
(d) 2
Find the least value of K so that 36 K36 is divisible
by 6.
2
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
(a) 1
(b) 6
(c) 2
(d) 3
Find the least value of K so that 2718 K6 is divisible
by 9.
(a) 9
(b) 3
(c) 2
(d) 8
What least value must be given to K so that the
number 39 K 0 is divisible by 11.
(a) 1
(b) 2
(c) 9
(d) 6
Find the sum of the first 50 natural numbers.
(a) 1275
(b) 1025
(c) 1235
(d) 1205
Find the sum of all natural numbers from 100 to
175.
(a) 10456
(b) 10452
(c) 10450
(d) 10455
Find the sum of all natural numbers between 100
and 175.
(a) 10450
(b) 10175
(c) 10170
(d) 10435
Find the sum of first 50 even numbers.
(a) 2550
(b) 2540
(c) 2050
(d) 2060
Find the sum of all the even numbers up to 300.
(a) 22650
(b) 22675
(c) 22600
(d) 22690
Find the sum of all even numbers from 100 to
175.
(a) 2218
(b) 5216
(c) 5206
(d) 5200
Find the sum of first 50 odd numbers.
(a) 1950
(b) 1800
(c) 2000
(d) 2500
Find the sum of all odd numbers between 100 and
175.
(a) 5069
(b) 5065
(c) 5070
(d) 5050
Find the sum of the squares of first 50 natural
numbers.
(a) 42925
(b) 42900
(c) 42860
(d) 42875
Find the sum of the cubes of first 10 natural
numbers.
(a) 3010
(b) 3040
(c) 3025
(d) 3020
Find the number of digits that are to be used in
numbering a book of 500 pages.
(a) 1392
(b) 1346
(c) 1325
(d) 1352
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Number System
37. Which of the following is divisible by 15 ?
(a) 3975
(b) 3575
(c) 3970
(d) 3580
38. What least number must be subtracted from 3475
to make it divisible by 50 ?
(a) 75
(b) 100
(c) 25
(d) 50
39. What least number must be added to 5718 to
make it exactly divisible by 40 ?
(a) 0
(b) 1
(c) 2
(d) 3
40. Find the least number by which 3900 be multiplied
to make it a perfect square.
(a) 3
(b) 13
(c) 39
(d) 56
41. How many numbers upto 800 are divisible by 24?
(a) 30
(b) 29
(c) 33
(d) 26
42. How many numbers upto 700 are divisible by both
3 and 5?
(a) 42
(b) 46
(c) 39
(d) 52
43. Find the number nearest to 2559 which is exactly
divisible by 35.
(a) 2535
(b) 2555
(c) 2540
(d) 2560
44. How many three digit numbers are divisible by
18 ?
(a) 55
(b) 50
(c) 52
(d) 56
45. How many integers from 3 to 30, inclusive are
odd ?
(a) 13
(b) 14
(c) 15
(d) 16
46. Sum of all prime numbers between 50 and 90 is
(a) 485
(b) 572
(c) 722
(d) 635
47. What is the unit digit in (476 × 198 × 359 × 242)?
(a) 8
(b) 6
(c) 4
(d) 2
48. For any natural number n, what is the value of
1
1
1


 
1.2 2.3 3.4

1
n  n  1
(a) 1
(b) > 1
(c) < 1
(d) Can’t be determined
49. Which is smallest prime number ?
(a) 0
(b) 1
(c) 2
(d) 3
50. Three natural numbers are in the ratio 2 : 3 : 4.
If the sum of squares of these numbers is 116,
then determine the numbers.
(a) 2, 3, 4
(b) 4, 6, 8
(c) 6, 9, 12
(d) 8, 12, 16
51. Find the number of divisors excluding 1 and itself
for the number 4225.
(a) 8
(b) 7
(c) 9
(d) 6
52. Find the number of prime factors of 211 × 75 ×
116 .
(a) 22
(b) 21
(c) 6
(d) 18
53. How many integers from 1 to 100 exist such that
each is divisible by 6 and also has 6 as a digit.
(a) 6
(b) 4
(c) 5
(d) 8
54. Sum of two numbers is 60 and their difference is
12. Find their product.
(a) 864
(b) 852
(c) 824
(d) 836
55. If 4/5 of a number is 36. Find 3 of the number.
(a) 27
(b) 25
(c) 22
(d) 21
56. If a piece of road is 3000 m and we have to supply
some lamp posts. One post is at each end and
distance between two consecutive lamp post is
75 m. Find the number of posts required.
(a) 41
(b) 39
(c) 40
(d) 36
57. The sum of the digits of a two digit number is 9.
If 9 is added to the number, then the digits are
reversed. Find the number.
(a) 36
(b) 63
(c) 45
(d) 54
58. If 2/7 of a number increased by 25 gives 45. Find
the number.
(a) 81
(b) 63
(c) 72
(d) 70
59. Out of four consecutive prime numbers, the
product of first three is 385 and the product of
the last three is 1001. Find the last number.
(a) 7
(b) 11
(c) 13
(d) 17
60. Three numbers are in the ratio 3 : 5 : 6. Sum of
the greatest and the smallest is equal to the sum
of the middle and 16. Find the smallest number.
(a) 12
(b) 20
(c) 24
(d) 16
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Number System
61. Sum of squares of two numbers is 60 and
difference of the squares is 12. Find the sum of
two numbers.
(a) 4
(b) 10
(c) 6
(d) 8
62. The product of two consecutive integers is 600.
Then, the sum of their squares is
(a) 1203
(b) 1206
(c) 1201
(d) 1204
63. Thrice a number diminished by 2 is 19. Find the
number.
(a) 5
(b) 7
(c) 4
(d) 6
64. The sum of 50 natural numbers is always divisible
by
(a) 17
(b) 3
(c) 5
(d) All of these
65. N = n; where n is natural number. The unit’s digit
of N can’t be
(a) 2
(b) 6
(c) 5
(d) 0
555
66. Given that 4A =333 + 555333, then A is divisible
by
(a) 2
(b) 3
(c) 37
(d) All of these
67. N = (a × b × c × d), where a, b, c, d are distinct
integers lying between –7 and 12, both inclusive.
The minimum value of N is
(a) –9240
(b) –840
(c) –2520
(d) None of these
68. The sum of two numbers is 85 and their difference
is 9. What is the difference of their squares?
(a) 765
(b) 845
(c) 565
(d) 645
69. Find the unit digit in the expansion of 32545
(a) 0
(b) 5
(c) 1
(d) 2
70. Find the unit digit in the expansion of 31448.
(a) 6
(b) 4
(c) 8
(d) 2
71. A worker was engaged for a certain number of
days and was promised to be paid ` 1755. He
remained absent for some days and was paid
` 1365 only. What were his daily wages ?
(a) ` 182
(b) ` 195
(c) ` 185
(d) ` 192
72. Which of the following cannot be a digit in the
unit’s place of a perfect square ?
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
(a) 7
(b) 1
(c) 5
(d) 0
The smallest number, which when added to the
sum of square of 9 to 10 gives a perfect square is
(a) 0
(b) 3
(c) 8
(d) 15
(9992 – 9982) is equal to
(a) 1
(b) 999
(c) 1997
(d) 998
A number when divided by 119 leaves the
remainder 19. If the same number is divided by
17, the remainder will be
(a) 19
(b) 10
(c) 7
(d) 2
If n is any positive integer, then 34n – 43n is
always divisible by
(a) 7
(b) 17
(c) 112
(d) 145
The number zero (0) is surrounded by the same
two digit number on both (left and right) sides:
for example 25025, 67067 etc. The largest number
that always divides such a number is
(a) 7
(b) 11
(c) 13
(d) 1001
The least number of five digits exactly divisible
by 456 is
(a) 10000
(b) 10012
(c) 10032
(d) 10056
Ashok had to do a multiplication. Instead of taking
35 as one of the multipliers, he took 53. As a result,
the product went up by 540. What is the new
product ?
(a) 1050
(b) 1590
(c) 1440
(d) None of these
Which of the following is a multiple of 88 ?
(a) 1392578
(b) 138204
(c) 1436280
(d) 143616
Which of the following fractions is less than 7/8
and greater than 1/3 ?
(a) 1/4
(b) 23/24
(c) 11/12
(d) 17/24
If the numbers : 169, 248, 416, 974, 612, 325 and
517 are arranged in descending order based on
the sum of the digits of each of these numbers,
the middle number will be
(a) 248
(b) 517
(c) 612
(d) 974
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Number System
83. Which of the following is true, if a = 3/4, b = 4/5
and c = 5/6?
(a) a < c < b
(b) a < b < c
(c) c < a < b
(d) b < a < c
84. In the array 48392874362754869364, the
number of instances where an even number is
followed by two odd numbers is
(a) 1
(b) 2
(c) 3
(d) 4
85. How many integers from 1 to 100 exist such that
each is divisible by 5 and also has 5 as a digit ?
(a) 10
(c) 12
(b) 11
(d) 20
86. If x is an even integer, which of the following is
an odd integer ?
(a) 3x + 2
(b) 7x
(c) 8x + 5
(d) x3
87. When the integer n is divided by 6, the remainder
is 3. Which of the following is not a multiple of 6?
(a) n – 3
(b) n + 3
(c) 2n
(d) 3n.
ANSWER
1. (c)
11. (a)
21. (b)
31. (c)
41. (c)
51. (b)
61. (b)
71. (b)
81. (d)
2.
12.
22.
32.
42.
52.
62.
72.
82.
(c)
(a)
(a)
(d)
(b)
(a)
(c)
(a)
(b)
3.
13.
23.
33.
43.
53.
63.
73.
83.
(a)
(a)
(d)
(a)
(b)
(c)
(b)
(d)
(b)
4.
14.
24.
34.
44.
54.
64.
74.
84.
(d)
(d)
(b)
(a)
(b)
(a)
(d)
(c)
(c)
5.
15.
25.
35.
45.
55.
65.
75.
85.
(b)
(b)
(d)
(c)
(b)
(a)
(c)
(d)
(b)
6.
16.
26.
36.
46.
56.
66.
76.
86.
(d)
(c)
(a)
(a)
(d)
(a)
(d)
(b)
(c)
7.
17.
27.
37.
47.
57.
67.
77.
87.
(d)
(b)
(c)
(a)
(c)
(c)
(a)
(d)
(d)
8.
18.
28.
38.
48.
58.
68.
78.
(c)
(d)
(b)
(c)
(c)
(d)
(a)
(c)
9.
19.
29.
39.
49.
59.
69.
79.
(d)
(d)
(a)
(c)
(c)
(c)
(b)
(b)
10.
20.
30.
40.
50.
60.
70.
80.
EXPLANATIONS

9 3  5  5 4  10
1.
 3  5  2  4  2
=
9
 2
 3
 5
 5
 4
 10
 8 2
=
18  2
16
= 
 15  4
19
2. 3x + (3x + 3) + (3x + 6) = 72
 9x + 9 = 72 = 9x = 72 – 9
45
135
45
35
x+y =
×y 
x =
y
100
100
100
100
 x : y = 35 : 45 = 7 : 9
7. Let the number be x. Then
63
=7
9
 The largest of them = 27.
3. Let the number be x.
or x =
55
1
 x = x  52
100
3
13
 x = 52  x = 240

60

1
1
2
x – 48 = x x  x = 48   x = 48
3
3
3
8.
2
x = 2 ´ 240 = 96
5
5
4. Let the number be 10x – y
x:y=2:3
(10y + x) – (10x + y) = 27
9y – 9x = 27  y – x = 3  y = x + 3
Putting this value of y, in (i)
x
2
=
 x=6 y=9
x 3
3
Hence the number is 69.
5. Apply the divisibility tears of 3 and 5.
6. Let the numbers be x and y
1
5
x = y
5
8
...(i)
35 + x = 4y ...(ii)
Solving (i) and (ii),
...(i)
35 
25
y = 4y 
8
y = 40
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(d)
(d)
(a)
(c)
(b)
(a)
(a)
(d)
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Number System
5 3 4

 x = 45
9. 
8 10 9
45 

9 10 8


= 540
4 3 5
10. Dividing 11158 by 70 we get 70 as remainder.
Thus 7 is to be added to make it divisible by 77.
11. We know that (an + 1) is always divisible by (a +
1) when n is odd.
12. Given that x 2 + y 2 + 2XY = 10 and
XY = 4
(Y + X)2 = X2 + Y2 + 2XY = 10 + 8 = 18

(Y + X) = 3 2 ...(i)
(Y – X)2 = X2 + Y2 – 2XY = 10 – 8 = 2

(Y – X) =
2
...(ii)
Sovling (i) and (ii), X =
2Y = 2 2 ,
X
1
=
Y
2
14. Clearly, for n = 237 (Q n is any positive integer)
the expression is divisible by 237. So it is false.
15. The number 899 lies between the squares of 29
and 30. Dividing 899 by the prime numbers less
than 30 we see that it is divisible by 29, so it is
not a prime number.
The number 899 has 4 factors 1, 29, 31 and 899.
16. According to the given condition the fraction is
( 2 + c)( 4 + c)
(5 +c)
Put 5 + c = t  c = t – 5
So, the fraction becomes
=
=
50  50  1
2
50  51
 1275
2
27. From 100 to 175 mean including both 100 and
175.
Sum of natural numbers upto 99
(t - 3)(t -1) t 2 - 4t + 3
=
t
t
99  100
= 4950
2
Sum of natural numbers upto 175
=
3 (t - 2)2 1
t - 4+ =
t
t
t
Hence the given expression is minimum if (t – 2)2
= 0 i.e., t = 2. Therefore c = –3 and for this value
the minimum value of the fraction is 
17. 13x = 180 + x
13x – x = 180 or, 12x = 180
180
or, x =
  x = 15
12
18. 10800 = 24 × 52 × 33
Number of divisors
= (4 + 1) (2 + 1) (3 + 1)
= 60.
19. Choosing 32 we see that the sum of the digits is 5
and when 9 is substracted from 32 the number
obtained is 23.
20. Let the number be x, x + 1, x + 2.
2x + 3(x + 1) + 4(x + 2) = 182
9x + 11 = 182x = 19
 The numbers are 19, 20, 21.
21. Place value = 5 × 1000 = 5000
Face value = 5
Product = 5000 × 5 = 25000
22. 3 + 9 + K + 2 + 0 = 14 + K
 Least value of K is 1.
23. Given number is divisible by 6, if it is divisible by
both 2 and 3.
 The least value of K is 3.
24. 2 + 7 + 1 + 8 + K + 6 = 24 + K
 Least value of K is 3.
25. 3 + K = 9 + 0
 K=6
26. Sum of the first 50 natural numbers
1
.
2
175  176
= 15400
2
 Sum of all natural numbers from 100 to 175
= 15400 – 4950 = 10450
28. Between 100 and 175 means excluding 100 and
175. Sum of natural numbers upto 100
=
=
100  101
 5050
2
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Number System
Sum of natural numbers upto 174
=
174  175
 15225
2
 Required sum = 15225 – 5050 = 10175
29. Sum of first 50 even numbers = 50(50 + 1) = 2550
300
= 150 even numbers Sum
2
of 150 even numbers = 150 (150 + 1) = 22650
30. Upto 300, there are
99  1
= 49 even numbers Sum
2
of 49 even numbers = 49 (49 + 1) = 2450 Upto
31. Upto 99, there are
175  1
= 87 even numbers Sum of
2
87 even numbers = 87 (87 + 1) = 7656
 Required sum = 7656 – 2450 = 5206
32. Sum of first 50 odd numbers = (50)2 = 2500
175, there are
100
= 50 odd numbers Sum
2
of first 50 odd numbers = (50)2 = 2500
33. Upto 100, there are
Upto 174, there are
174
= 87 odd numbers
2
Sum of first 87 odd numbers = (87)2 = 7569
 Required sum = 7569 – 2500 = 5069
34. Sum of squares of first 50 natural numbers
=
=
50  50  1  2  50  1
50  51  101
= 42925
6
35. Sum of cubes of first 10 natural numbers
2
 10  11 
 
2 

43.
35)2559(73
245
109
105
4
The two numbers nearest to 2559 divisible by 35
are
(a) 2559 – 4 = 2555
(b) 2559 + (35 – 4) = 2590
 The required number is 2555.
44. Three digit numbers are from 100 to 999.
Upto 100, there are
6
 10  10  1  

= 
2


37. A number is divisible by 15, if it is both divisible
by 3 and 5. 3975 is the only number divisible by
both 3 and 5.
38. Remainder when 3475 is divided by 50 is 25.
 Least number that should be subtracted = 25
39. Remainder when 5718 is divided by 40 is 38.
 Least number that should be added = 40 –38
=2
40. 3900 = 2 × 2 × 5 × 5 × 3 × 13
 The least number by which 3900 be multiplied
to make it a perfect square = 3 × 13 = 39
41. Quotient when 800 is divided by 24 is 33. There
are 33 numbers upto 800 divisible by 24.
42. Quotient when 500 is divided by the LCM of 3
and 5 ie, 15 is 46.
 There are 46 numbers upto 700.
2
= 552 = 3025
36. One digit numbers from 10 to 9 = 9
Two digit numbers from 100 to 99 = 90
Three digit numbers from 100 to 500 = 401
 Total number of digits that are to be used
= (9 × 1) + (90 × 2) + (401 × 3)
= 9 + 180 + 1203
= 1392
100
= 5 numbers
18
Upto 999, there are 999 = 55 numbers
Required number = 55 – 5 = 50 numbers
45. Total number of numbers between 3 to 30 inclusive
= 28
Out of which number of odd numbers
= number of even numbers
=
28
 14
2
46. Required sum
= 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89
= 635
47. Unit digit in 476 × 198 × 359 × 242
= unit digit in 6 × 8 × 9 × 2
= unit digit in 864 = 4
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Number System
48. Given expression
1
1
1


 
1.2 2.3 3.4

59. Let the four numbers be p, q, r and s.
pqr = 385, qrs = 1001
1
n  n  1
1  1 1  1 1





 
=  1
2   2 3   3 4 

qrs 1001 13


pqr 385
5
1 
 1
  
 1 
n
n

1
n

 1
n 1 n 1
  Denominator is greater than numerator.)
49. 2 is smallest prime number.
50. Let the required numbers be 2x, 3x and 4x. Then,
(2x)2 + (3x)2 + (4x)2 = 116
29x2 = 116
x2 = 4  x = 2
= 1
51.
52.
53.
54.
55.
The numbers, are 4, 6, 8.
4225 = 52 × 132
Total divisor = (2 + 1) (2 + 1) = 3 × 3 = 9
The number of divisors excluding 1 and itself
=9–2=7
Number of prime factors = 11 + 5 + 6 = 22
The required numbers = 6, 36, 60, 66, 96 = 5
numbers
Let the 2 numbers be x and y.
Solving, x = 24, y = 36
Product = 24 × 36 = 864
Let the number be x.
4
x  36 
5
60.
61.
62.
63.
64.
x = 45
3
3
x
 45  27
5
5
56. Number of lamp posts required
Now,
65.
3000
 1 = 40 + 1 = 41
=
75
57. Let the units place digit be x and tens place digit
be y.
x+y=9
...(i)
(l0y + x) + 9 = (y + 10x)
... (ii)
Solving, x = 5 and y = 4.
 The required number is 45.
58. Let the number be x.
2
x  25 = 45. Solving, x = 70
7
66.
67.
68.
s 13

p 5
So, p = 5 and s = 13.
 Required number is 13.
Let the numbers be 3x, 5x and 6x.
Now,
3x + 6x = 5x + 16
x =4
 The smallest number is 3x = 12.
Let the number be x and y.
x2 + y2 = 60, x2 – y2 = 12
Adding,
2x 2 = 72
x = 6 and y = 4
 Required sum is 10.
Let the two numbers be x and x + 1.
x (x + 1) = 600
Solving x = 24, so the numbers are 24 and 25.
The sum of their squares is = 242 + 252 = 1201
Let the number be x.
3x – 2 = 19
Solving
x=7
 The required number is 7.
Sum of first 50 natural numbers
50  51
= 25x51
2
which is divisible by 3, 5, 17.
Since, n ! is always even for n > 1. Therefore, 5
cannot be the unit digit.
Since, 4A = 333555 + 555333 is a very large number
so taking a similar smaller exponent. (Here, the
exponent is odd which is 1.)
  4A = 3331 + 5551  4A = 888  A = 222
which is divisible by 2, 3 and 37.
Take, –7, 12, 11 and 10 because these will give
minimum value.
x + y = 85
... (i)
x–y=9
... (ii)
On multiplying Eqs. (i) and (ii)
(x + y) (x – y) = 85 × 9
x2 – y2 = 765
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Number System
69. The units digit in 32545 is the same as that of 545
Now, 51 = 5,52 = 25,53 = 125, 54 = 625, …
We see that, the units digit is always 5.
 The units digit in 32545 is 5.
70. The units digits in the expansion of 31448 is the
same as that of 448.
Now, 41 = 4, 42 =16,43 = 64,44 = 256.
We see that, the units digit in 41 is the same as
that in 43.
So, the units digit is 448 is the same as is 448 = 6.
71. The daily wages of the worker is the HCF of 1755
and 1365. HCF of 1755 and 1365 is 195.
 The daily wages of the worker is `195.
72. Perfect squares are
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, …
Clearly, 7 cannot be at unit’s digit.
73. 92 + 102 = 81 + 100 =181, 196 is the nearest
perfect square. Hence, 15 should be added.
74. 9992 – 9982 = (999 + 998) (999 – 998)
= (1997) (1) = 1997
75. The number is = 119n + 19 = 138,257
When these numbers are divided by 17, then
remainder is 2.
76. 34(1) – 43(1) = 81 – 64 = 17,
34(2) – 43(2) = 812 – 642 = 2465
It is always divisible by 17.
77. The largest such number is 1001.
78. The least five digit number is 10000. When it is
divided by 456 the remainder is 424.
So, the least five digit number that is exactly
divisible by 456 = 10000 + 456 – 424 = 10032.
79. Suppose, one of the multiplier is x, then
53x – 35x = 540
18x = 540
x =30
 New multiplication = 30 × 53 = 1590
80. Only 143616 is a multiple of both 11 and 8 so it is
a multiple of 88.
7
23
= 0.875, 3 = 0.333, 1 = 0.25,
= 0.958
8
24
7
1
 0.875,  0.333,
8
3
1
23
 0.25,
 0.958
4
24
11
17
 0.916,
 0.708
12
24
Clearly, 0.708 lies between 0.333 and 0.875.
 Required fraction is 17/24.
82.
1 + 6 + 9 = 16 ... (ii)
2 + 4 + 8 = 14 ...(iii)
4 + 1 + 6 = 11 ...(v)
9 + 7 + 4 = 20 ...(vi)
6 + 1 + 2 = 9 ...(vii)
3 + 2 + 5 = 10 ...(vi)
5 + 1 + 7 = 13 ...(iv)
 The middle number will be 517.
81.
83.
a=
3
 0.75
4
b=
4
 0.800
5
5
 0.833
6

a<b<c
In the array 48392874362754869364; 8, 2 and 6
ie, three number of required instances.
These numbers are following
5, 15, 25, 35, 45, 50, 55, 65, 75, 85, 95
So, total 11 such type of numbers.
If x is an even integer, then clearly 8x will also be
an even integer.
So, 8x + 5 will be an odd integer.
If n is divided by 6 and the remainder is 3, then n
– 3 and n + 3 are divisible by 6.
2n is also divisible by 6.
But, 3n is not divisible by 6.
‘
c=
84.
85.
86.
87.
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