Notes / Rough Work
NUMBER SYSTEM &
WORKING WITH NUMBERS
System of numbers Notes / Rough Work
With the help of a tree diagram, numbers can be classified as follows
Numbers
Real Imaginary
Rational Irrational (2,3 etc.)
Integer Fraction Algebraic Transcendental
(Roots of poly) (e, etc.)
Proper Improper Mixed
1 1
(1/3, 2/5, 3/8) (4/3, 5/2, 6/5 etc.) (1 /4, 3 /15, ... etc.)
Whole Number Natural Negative odd Even
(0, 1, 2, .....) (±1, ±3, ±5 ...) (multiples of 2)
Prime Composite Perfect
(2, 3, 5, 7, ....) (4, 6, 8, ...) (6, 28, ....)
Real numbers
Real numbers are those which can represent actual physical quantities e.g.
temperature, length, height etc. Real numbers can also be defined as numbers that
can be represented on the number line.
0.6 2
-2.3
-3 -2 -1 0 1 2 3
(1) of (36)
,Natural numbers (positive integers)
Notes / Rough Work
These are the counting numbers used to count physical quantities. e.g. 1, 2, 3, ...,
105, ..., 326, 15957, ..., 37950046, ... . The set of natural numbers is denoted by
N = {1, 2, 3, 4, 5, 6, 7 ..... }
E1. S and L are the smallest and the largest n–digit natural numbers respectively. L – S
is always divisible by
(1) 9 (2) 10
(3) 9 and 10 (4) None of these
Sol. When n = 1, L – S = 8, when n = 2, L – S = 89 etc. Thus we see that L – S is never
divisible by 9 or 10.
E2. If ab is two digit number and 7b + a = 23k, where k is a natural number. The
largest number that always divides the product of ab and twice of ab is
(1) 1058 (2) 46
(3) 92 (4) None of these
Sol. We have 7b + a = 23k or, a = 23k – 7b. Now the value of the number ab is
10a + b = 10(23k – 7b) + b= 230k – 69b = 23(10k – 3b) ab is a multiple of 23.
Product of ab and 2ab will always be divisible by 23 × 23 × 2 = 1058
Whole numbers
The numbers 0, 1, 2, 3, 4, ... are whole numbers.
The set of all non-negative integers (i.e. zero + natural numbers) is said to be the set
of Whole Numbers and is denoted as W = {0, 1, 2, 3, ... }.
E3. abc is a three digit whole number so that abc = a 3 + b 3 + c 3 . 300 abc 400.
What is the value of a + b + c?
(1) 10 (2) 11
(3) 12 (4) Data insufficient
Sol. There are two such numbers in the given range viz 370 and 371. Hence, (4).
E4. If 17 is added to product of two consecutive whole numbers we always get a /an
(1) Prime number (2) Even number
(3) Odd number (4) None of these
Sol. You must have answered (1) thinking
1 × 2 + 17 = 19, 2 × 3 + 17 = 23, 3 × 4 + 17 = 37 etc.
but if we take 16 × 17 + 17 = 289. It is a composite number. Out of two consecutive
whole numbers one will always be even number and so their product will be even.
When an odd number 17 is added to an even number we always get an odd
number. Hence, (3).
Integers
The set of all natural numbers (positive, zero, negative) are together known as integers.
The set of integers is denoted as I where I = {0, ±1, ±2, ±3, ... }.
Zero and positive integers are called as non-negative integers.
E5. A is the smallest integer that when multiplied with 3 gives a number made of
5’s only. Sum of the digits of A is B. Sum of the digits of B is C. What is the
value of C 3 ?
(1) 125 (2) 64
(3) 216 (4) None of these
Sol. A = 185, B = 14, C = 5. Hence, (1).
(2) of (36)
,E6. P is integer. P > 883. If P > 883. If P – 7 is a multiple of 11 then the largest
number that will always divide (P + 4) (P + 15) is Notes / Rough Work
(1) 11 (2) 121
(3) 242 (4) None of these
Sol. If (P – 7) is a multiple of 11, (P + 4) and (P + 15) must be multiple of 11 as well
because P + 4 = (P – 7) + 11 and P + 15 = (P – 7) + 22. Since (P + 4) and (p + 15)
are consecutive multiples of 11, so one of them must be an even number.
Hence, (P + 4) (P + 15) will always be divisible by 11 × 11 × 2 = 242. Hence, (3).
E7. Vijay writes all the numbers from 100 to 999. The number of zeroes that he uses
is m, the number of fives that he uses is n and the number of 8’s that he uses is
p. What is the value of n + p – m?
(1) 280 (2) 380
(3) 180 (4) None of these
Sol. m = 180, n =280, p = 280, Hence, (2).
Rational numbers
p
p and q (q 0) are integers. Then is known as a rational number. Thus the set Q of
q
the rational numbers is given by
Q
RS p : p, q I and q 0
UV
Tq W
7 23 2
Naturally, fractions such as , , are called rational numbers. This definition
9 16 5
also emphasises that any integer can also be a rational number since p = p/1, p I.
Any positive rational number p/q, after actual division, if necessary can be expressed
as,
p r
m where m is non-negative integer and 0 r < q
q q
41 1 3 3 10 0
For example, 8 ; 0 ; 10 10 .
5 5 5 5 1 1
For the decimal representation of a fraction p/q, we have merely to consider the
decimal form of fraction r/q which we usually write to the right of the decimal point.
Consider some fractions given below.
(1) 1/2 = 0.5 (2) 3/5 = 0.6
(3) 1/4 = 0.25 (4) 1/5 = 0.2
(5) 1/8 = 0.125 (6) 1/6 = 0.1666...
(7) 5/11 = 0.4545... (8) 1/3 = 0.33...
(9) 7/12 = 0.583333...
Note that the dots ........ represent endless recurrence of digits.
Examples (1), (2), (3), (4) and (5) suggest that we have decimal form of the
‘terminating type’. While examples (6), (7), (8) and (9) tell us that we have decimal
form of the ‘non-terminating type’.
(3) of (36)
, In case of ‘non-terminating type’ we have decimal fractions having an infinite number
of digits. Some decimal fractions from this group have digits repeating infinitely. They Notes / Rough Work
are called ‘repeating or recurring’ decimals.
In ‘endless recurring or infinite repeating’ decimal fractions we can see that when p is
actually divided by q the possible remainders are 1, 2, 3, ..... , q - 1. So one of them
has to repeat itself in q steps. Thereafter the earlier numeral or group of numerals
must repeat itself.
Note
(1) All the rational numbers thus can be represented as a finite decimal (terminating
type) or as a recurring decimal.
(2) The recurring digits from the recurring group are indicated by putting a dot above
the first and last of them or a bar above the recurring group.
For example
(i) 0.333 ....... as 0. 3 or 0 . 3 (ii) 1.2555 .... as 1.2 5 or 1 . 25
(iii) 3.142142142 ..... as 3. 142 or 3 .142
Every infinite repeating decimal can be expressed as a fraction.
Irrational numbers
Each non-terminating recurring decimal is a rational number. Thus the number which is
a non-terminating non recurring decimal or more simply the number which can not be
written as fraction (i.e. in the form p/q), is called an irrational number.
E.g. 2 = 1.414213562373095.....
= 3.141592653589793.....
log 2= 0.301029995663981..... etc.
Prime numbers
A positive integer which is not equal to 1 and is divisible by itself and 1 only is called a
prime number.
Ex. 2, 3, 5, 7, 11, 13, 17, 19 etc.
Thus, for the prime number 131 there are no factors besides 131 and 1.
E8. P is a prime number greater than 5. What is the remainder when P is divided by 6?
(1) 5 (2) 1
(3) 1 or 5 (4) None of these
IDENTIFYING A PRIME
Sol. Any prime number greater than 3 is of the form 6k 1 so when it is divided by 6
NUMBER
the remainder will obviously be 1 or 5.
E9. The average of three prime no’s is 223/3. What is the difference between the If a number has no prime
greatest and the smallest number? factor upto its square root,
(1) 8 (2) 16 it is prime. For example, let’s
(3) Data inadequate (4) None of these check 257. Now 257 16,
so we check all prime
Sol. We can have at least two such sets of prime numbers 71, 73, 79 and 67, 73, 83. numbers upto 16, i.e., 2, 3,
Hence, (3). 7, 11, 13. As no number
divides 257, it is prime.
(4) of (36)
NUMBER SYSTEM &
WORKING WITH NUMBERS
System of numbers Notes / Rough Work
With the help of a tree diagram, numbers can be classified as follows
Numbers
Real Imaginary
Rational Irrational (2,3 etc.)
Integer Fraction Algebraic Transcendental
(Roots of poly) (e, etc.)
Proper Improper Mixed
1 1
(1/3, 2/5, 3/8) (4/3, 5/2, 6/5 etc.) (1 /4, 3 /15, ... etc.)
Whole Number Natural Negative odd Even
(0, 1, 2, .....) (±1, ±3, ±5 ...) (multiples of 2)
Prime Composite Perfect
(2, 3, 5, 7, ....) (4, 6, 8, ...) (6, 28, ....)
Real numbers
Real numbers are those which can represent actual physical quantities e.g.
temperature, length, height etc. Real numbers can also be defined as numbers that
can be represented on the number line.
0.6 2
-2.3
-3 -2 -1 0 1 2 3
(1) of (36)
,Natural numbers (positive integers)
Notes / Rough Work
These are the counting numbers used to count physical quantities. e.g. 1, 2, 3, ...,
105, ..., 326, 15957, ..., 37950046, ... . The set of natural numbers is denoted by
N = {1, 2, 3, 4, 5, 6, 7 ..... }
E1. S and L are the smallest and the largest n–digit natural numbers respectively. L – S
is always divisible by
(1) 9 (2) 10
(3) 9 and 10 (4) None of these
Sol. When n = 1, L – S = 8, when n = 2, L – S = 89 etc. Thus we see that L – S is never
divisible by 9 or 10.
E2. If ab is two digit number and 7b + a = 23k, where k is a natural number. The
largest number that always divides the product of ab and twice of ab is
(1) 1058 (2) 46
(3) 92 (4) None of these
Sol. We have 7b + a = 23k or, a = 23k – 7b. Now the value of the number ab is
10a + b = 10(23k – 7b) + b= 230k – 69b = 23(10k – 3b) ab is a multiple of 23.
Product of ab and 2ab will always be divisible by 23 × 23 × 2 = 1058
Whole numbers
The numbers 0, 1, 2, 3, 4, ... are whole numbers.
The set of all non-negative integers (i.e. zero + natural numbers) is said to be the set
of Whole Numbers and is denoted as W = {0, 1, 2, 3, ... }.
E3. abc is a three digit whole number so that abc = a 3 + b 3 + c 3 . 300 abc 400.
What is the value of a + b + c?
(1) 10 (2) 11
(3) 12 (4) Data insufficient
Sol. There are two such numbers in the given range viz 370 and 371. Hence, (4).
E4. If 17 is added to product of two consecutive whole numbers we always get a /an
(1) Prime number (2) Even number
(3) Odd number (4) None of these
Sol. You must have answered (1) thinking
1 × 2 + 17 = 19, 2 × 3 + 17 = 23, 3 × 4 + 17 = 37 etc.
but if we take 16 × 17 + 17 = 289. It is a composite number. Out of two consecutive
whole numbers one will always be even number and so their product will be even.
When an odd number 17 is added to an even number we always get an odd
number. Hence, (3).
Integers
The set of all natural numbers (positive, zero, negative) are together known as integers.
The set of integers is denoted as I where I = {0, ±1, ±2, ±3, ... }.
Zero and positive integers are called as non-negative integers.
E5. A is the smallest integer that when multiplied with 3 gives a number made of
5’s only. Sum of the digits of A is B. Sum of the digits of B is C. What is the
value of C 3 ?
(1) 125 (2) 64
(3) 216 (4) None of these
Sol. A = 185, B = 14, C = 5. Hence, (1).
(2) of (36)
,E6. P is integer. P > 883. If P > 883. If P – 7 is a multiple of 11 then the largest
number that will always divide (P + 4) (P + 15) is Notes / Rough Work
(1) 11 (2) 121
(3) 242 (4) None of these
Sol. If (P – 7) is a multiple of 11, (P + 4) and (P + 15) must be multiple of 11 as well
because P + 4 = (P – 7) + 11 and P + 15 = (P – 7) + 22. Since (P + 4) and (p + 15)
are consecutive multiples of 11, so one of them must be an even number.
Hence, (P + 4) (P + 15) will always be divisible by 11 × 11 × 2 = 242. Hence, (3).
E7. Vijay writes all the numbers from 100 to 999. The number of zeroes that he uses
is m, the number of fives that he uses is n and the number of 8’s that he uses is
p. What is the value of n + p – m?
(1) 280 (2) 380
(3) 180 (4) None of these
Sol. m = 180, n =280, p = 280, Hence, (2).
Rational numbers
p
p and q (q 0) are integers. Then is known as a rational number. Thus the set Q of
q
the rational numbers is given by
Q
RS p : p, q I and q 0
UV
Tq W
7 23 2
Naturally, fractions such as , , are called rational numbers. This definition
9 16 5
also emphasises that any integer can also be a rational number since p = p/1, p I.
Any positive rational number p/q, after actual division, if necessary can be expressed
as,
p r
m where m is non-negative integer and 0 r < q
q q
41 1 3 3 10 0
For example, 8 ; 0 ; 10 10 .
5 5 5 5 1 1
For the decimal representation of a fraction p/q, we have merely to consider the
decimal form of fraction r/q which we usually write to the right of the decimal point.
Consider some fractions given below.
(1) 1/2 = 0.5 (2) 3/5 = 0.6
(3) 1/4 = 0.25 (4) 1/5 = 0.2
(5) 1/8 = 0.125 (6) 1/6 = 0.1666...
(7) 5/11 = 0.4545... (8) 1/3 = 0.33...
(9) 7/12 = 0.583333...
Note that the dots ........ represent endless recurrence of digits.
Examples (1), (2), (3), (4) and (5) suggest that we have decimal form of the
‘terminating type’. While examples (6), (7), (8) and (9) tell us that we have decimal
form of the ‘non-terminating type’.
(3) of (36)
, In case of ‘non-terminating type’ we have decimal fractions having an infinite number
of digits. Some decimal fractions from this group have digits repeating infinitely. They Notes / Rough Work
are called ‘repeating or recurring’ decimals.
In ‘endless recurring or infinite repeating’ decimal fractions we can see that when p is
actually divided by q the possible remainders are 1, 2, 3, ..... , q - 1. So one of them
has to repeat itself in q steps. Thereafter the earlier numeral or group of numerals
must repeat itself.
Note
(1) All the rational numbers thus can be represented as a finite decimal (terminating
type) or as a recurring decimal.
(2) The recurring digits from the recurring group are indicated by putting a dot above
the first and last of them or a bar above the recurring group.
For example
(i) 0.333 ....... as 0. 3 or 0 . 3 (ii) 1.2555 .... as 1.2 5 or 1 . 25
(iii) 3.142142142 ..... as 3. 142 or 3 .142
Every infinite repeating decimal can be expressed as a fraction.
Irrational numbers
Each non-terminating recurring decimal is a rational number. Thus the number which is
a non-terminating non recurring decimal or more simply the number which can not be
written as fraction (i.e. in the form p/q), is called an irrational number.
E.g. 2 = 1.414213562373095.....
= 3.141592653589793.....
log 2= 0.301029995663981..... etc.
Prime numbers
A positive integer which is not equal to 1 and is divisible by itself and 1 only is called a
prime number.
Ex. 2, 3, 5, 7, 11, 13, 17, 19 etc.
Thus, for the prime number 131 there are no factors besides 131 and 1.
E8. P is a prime number greater than 5. What is the remainder when P is divided by 6?
(1) 5 (2) 1
(3) 1 or 5 (4) None of these
IDENTIFYING A PRIME
Sol. Any prime number greater than 3 is of the form 6k 1 so when it is divided by 6
NUMBER
the remainder will obviously be 1 or 5.
E9. The average of three prime no’s is 223/3. What is the difference between the If a number has no prime
greatest and the smallest number? factor upto its square root,
(1) 8 (2) 16 it is prime. For example, let’s
(3) Data inadequate (4) None of these check 257. Now 257 16,
so we check all prime
Sol. We can have at least two such sets of prime numbers 71, 73, 79 and 67, 73, 83. numbers upto 16, i.e., 2, 3,
Hence, (3). 7, 11, 13. As no number
divides 257, it is prime.
(4) of (36)
