In this lecture sheet, the topics covered are:

Content Page
Types of Number 2
Introduction to Fractions 3
Properties of Number Line 3
Positive Numbers, Negative Numbers and Zero 5
Odd and Even Numbers 6
Divisibility Test 8
Factors and Multiples 9
HCF and LCM 10
Prime Numbers and Co-primes 10
Numbers
The most common number system that people use is the base ten number system. The base ten
number system is also called the decimal number system. A symbol for a number is made up of these
ten digits. The position of the digits shows how big the number is. For example, the number 23 in the
decimal number system really means (2 times 10) plus 3, and 101 means 1 times a hundred (=100)
plus 0 times 10 (=0) plus 1 times 1 (=1).

Types of Numbers

Real Number

A real number is a value that represents any quantity along a number line. Because they lie on a
number line, their size can be compared. You can say one is greater or less than another, and do
arithmetic with them. In mathematics, a real number is a value of a continuous quantity that can
represent a distance along a line.

Real numbers can be thought of as points on an infinitely long line called the number line or real line,
where the points corresponding to integers are equally spaced. It can be extended infinitely in any
direction and is usually represented horizontally. The numbers on the number line increase as one
moves from left to right and decrease on moving from right to left.

Real numbers are divided into two categories:

● Rational numbers
● Irrational numbers

Rational Number
A rational number is a number that can be in the form p/q where p and q are integers and q is not
equal to zero. For example -

Just remember!
“q” cannot be equal to zero

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Irrational Number
The opposite of rational numbers are irrational numbers. In simple terms, irrational numbers are real
numbers that can’t be written as a simple fraction like p/q.
Take π.
π is a real number. But it’s also an irrational number, because you can’t write π as a simple fraction:
π = 3.1415926535897932384626433832795 (and counting)
There’s no way to write π as a simple fraction, so it’s irrational.
The same goes for √2.
The √2 equals 1.4142135623730950... (etc).
You can’t make √2 into a simple fraction, so it’s an irrational number.

Integers
An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Examples
of integers are: -5, 1, 5, 8, 97, and 3043. Examples of numbers that are not integers are: -1.43, 1 3/4,
3.14, 0.09, and 5643.1.

Introduction to Fractions
A fraction is expressed as one number over another number. The number on top is called the
numerator and the number at the bottom is called the denominator. Fractions are of three types:

● Proper fractions:
A fraction where the numerator (the top number) is less than the denominator (the bottom
number). Example: 1/4 (one quarter) and 5/6 (five sixths) are proper fractions. The
absolute value of proper fractions are greater than zero and smaller than 1.
So, if x is 5/6 of y, then y>x.
● Improper fraction:
A fraction where the numerator (the top number) is greater than or equal to the
denominator (the bottom number). So it is usually "top-heavy". Example: 5/3 (five thirds)
and 9/8 (nine eighths) are improper fractions. The absolute value of improper fractions
are more than 1.
So, if x is 9/8 of y, then x>y.
● Mixed fraction:
A whole number and a fraction combined into one "mixed" fraction. Example: 1½ (one and
a half) is a mixed fraction. Here 1 is and integer and ½ is a fraction. 1+½ = 3/2. So mixed
fraction is a different way of representing improper fractions. Like improper fractions, the
absolute value of a mixed fraction is always more than 1.

Note: We get the integer part of the improper fraction by dividing the numerator by the denominator and
discarding the indiscrete part. For example, in case of 15/4, we know that 4 goes into 15 three times
𝟑
with 3 as left over or reminder. So, 15/4 = 3 𝟒. This converts the improper fraction into a mixed fraction.

Properties of Number Line

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A number line is a picture of a graduated straight line that serves as abstraction for real numbers,
denoted by R. Every point of a number line is assumed to correspond to a real number, and every real
number to a point.

To draw a number line we draw a line with several vertical dashes in it and ordered numbers below
the line, both positive and negative. The number corresponding to the point on the number line is
called the coordinate of the number line.

Properties of numbers between zero and one

➢ 0.1, 0.5, 0.66 etc are numbers between 0 and 1

➢ Any number between zero and one, when raised to the power of an integer greater than one,
becomes smaller than the original number i.e if 0<x<1, x2 < x.
2
For example: (0.1) = 0.01 and 0.01 < 0.1.
➢ Any number between zero and one, when represented as fraction, is always a proper
fraction.

Example:

● If x is greater than zero, but less than 1, which of the following is the largest?

A. x4 B. x3 C. x D. 1/x4 e. 1/x3

Answer: (D)

Explanation:
Any number between zero and one, when raised to the power of an integer greater than one,
becomes smaller than the original number. So, x4 is the smallest among x4, x3, x2 and x. So the
reciprocal of x4 will be the greatest.

Properties of numbers between -1 and 0:

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  -0.3, -0.2, -0.88 etc are numbers between -1 and 0.
➢ If the index of any negative number is an even integer, it becomes a positive number. So the
number becomes greater.
➢ If a number between -1 and zero, is raised to a power of an odd number, the numerical value
increases. For example, (-0.5)3 = (-0.125) and (-0.125) > (-0.5)

Example:

● If X is greater than -1, but less than 0, which of the following is the largest?

A. x4 B. x3 C. x D. 1/x4 E. 1/x3

Answer: (D)

Explanation:
If the index of any negative number is an even integer, it becomes a positive number. So the number
becomes greater. So, x4and x2 are positive. And x4< x2. So, the reciprocal of x4 will be the greatest.

Positive numbers, negative numbers and zero

Positive Numbers:

Positive numbers are numbers that are greater than zero.

Numbers can be positive, negative or zero. Zero is neither positive nor negative. Positive numbers
are the ones you most encounter in everyday life, such as 34, 9.22, etc. When shown on a number
line, they are the ones drawn on the right of zero, getting larger as you move to the right.

Zero:

Zero is considered the middle point of the number line. Measurements one way are positive, and they
are negative the other way.

Zero is neither positive nor negative. It is considered a Non-negative number.

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 Negative Numbers:

Negative numbers are numbers that are less than zero.

When you first encounter negative numbers they can be perplexing. How can a bowl contain less
than zero oranges? In fact, the counting numbers cannot be negative for this reason. It makes no
sense. But when we use scalar numbers, we are measuring something like temperature or height,
negative values are useful.

Odd and Even numbers:

Odd Numbers:

An odd number is an integer (never a fraction) which is not a multiple of two. If it is divided by two, it
leaves a remainder or the result can be represented as a fraction. They are generally represented as
“2x + 1” or ” 2x – 1” where x is an integer.

Even Number:

Any integer (never a fraction) that can be divided exactly by 2. The last digit of any even number is
always 0, 2, 4, 6 or 8. They are generally represented as “2 x “ where x is an integer.

Properties of odd and even numbers:

Addition and Subtraction:

Operation Result Example

Even + Even Even 2+4=6

Even + Odd Odd 6+3=9

Odd + Even Odd 5 + 12 = 17

Odd + Odd Even 3+5=8

(The same happens when subtracting is done instead of adding.)

Multiplication:

Operation Result Example

Even × Even Even 4 × 8 = 32

Even × Odd Even 4 × 7 = 28

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Odd × Even Even 5 × 8 = 40

Odd × Odd Odd 5 × 7 = 35

Exponents:

Operation Result Example

Even ^ Even Even 2^2=4

Even ^ Odd Even 2^3=8

Odd ^ Even Odd 3^2=9

Odd ^ Odd Odd 3 ^3 = 27

Example:

● If x, y and z are consecutive positive integers and if x>y>z, which of the following must be a
positive odd integer?
A. xyz B. (x–y) (y–z) C. x–yz D. x(y+z) E. none of these

Answer: option (B)

Explanation:
Since, all the variables are consecutive positive integers, x must be either odd or even.
Let, x be an odd number. If x, y and z are 3, 2 and 1 respectively
Option (a) 3 x 2 x 1 = 6 which is not a positive odd integer, so it can be eliminated.
Option (b) (3-2) x (2-1) = 1 which is a positive odd integer
Let, x be an even number. If x, y and z are 4, 3 and 2 respectively
Option (b) (4-3) x (3-2) = 1 which is a positive odd integer
Option (c) 4 – (3x2) = -2 which is not a positive integer, so it can be eliminated
Option (d) 4 x (3+2) = 10 which is a not a positive integer, so it can be eliminated.
So, option (b) is the correct answer

● The average of five consecutive odd positive integers is always

A. an odd number B. divisible by 5 C. divisible by 8

D. both (B) & (C). E. either (A) & (B)

Answer: option (A)

Explanation:
The sum of ‘n’ consecutive odd integers is 2nx+ n2. (Where x is an integer.)
The sum of five consecutive odd numbers is (2x+1) + (2x+3) + (2x+5) + (2x+7) + (2x+9) =
10x+25 = 5(2x+5)
Therefore the average of five consecutive odd positive integer is, (10x+25)/5 = 2x+5
2x is an even integer, so 2x+5 is an odd integer. Therefore 2x+5 is an odd number.

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 ● The average of five consecutive even positive integers is always

A. an odd number B. divisible by 5 C. divided by 8

D. both (B) & (C). E. either (B) & (A)

Answer: option (A)

Explanation:
The sum of ‘n’ consecutive even integers is 2nx + n(n-1). (Where x is an integer.)
The sum of five consecutive odd numbers is 2x+(2x+2)+(2x+4)+(2x+6)+(2x+8) = 10x+5(5-1)
= 10x+20 = 10(x+2).
Therefore the average of five consecutive even positive integer is, (10x+20)/5 = 2(x+2), which
is always an even number.

Divisibility Test
A divisibility test is a rule for determining whether one whole number is divisible by another. It is a
quick way to find factors of large numbers.

Some common Divisibility test:

● A number is divisible by 2 if it is an even number.

● A number is divisible by 3 if the sum of the digits is divisible by 3. For example, 534. Here, 5 +
3 + 4 = 12. Here, 12 is divisible by 3. So, the number 534 is divisible by 3.
● A number is divisible by 4 if the number formed by the last two digits is divisible by 4. For
example, 5240. Here, the last two digits form 40, which is divisible by 4. So, the entire
number 5240 is divisible by 4.
● A number is divisible by 5 if the unit’s digit is either 5 or 0. For example, 520, 325 etc.
● A number is divisible by 6 if the number is divisible by both 2 and 3. For example, 894. Here,
It’s an even number. So, it’s divisible by 2. Again, 8+9+4 = 21. 21 is divisible by 3. So, 894 is
also divisible by 3 as well. Thus, we can conclude 894 is divisible by 6.
● A number is divisible by 8 if the number formed by the last three digits is divisible by 8. Take
for instance, the number 523848. The number formed by the last three digits is 848. 848 is
divisible by 8. So the number 523848 is divisible by 8.
● A number is divisible by 9 if the sum of the digits is divisible by 9. For example, the sum of
the digits of the number 1584 is 18, which is divisible by 9. So the number 1584 is divisible
by 9.
● A number is divisible by 10 if the unit’s digit is 0. For example, the unit digit of the number
1580, is 0. So the number is divisible by 10.
● 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 should be zero or a multiple of 11. For example the sum of
the digits at the odd places of the number 148965 is 1+8+6=15. And the sum of the digits at
the even places is 4+9+5=18. The difference of 18 and 15 is 3 which is not divisible by 11. So
the number 148965 is not divisible by 11.
● A number is divisible by 12, when it is divisible by both 3 and 4
● A number is divisible by 24, when it is divisible by both 3 and 8
● A number is divisible by 36, when it is divisible by both 4 and 9
● A number is divisible by 25, when the last two digits are 00 or divisible by 25
● A number is divisible by 125, if the last three digits are 000 or divisible by 125
● A number is divisible by 27, if the sum of the digits of the number is divisible by 27

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

Factor and Multiples

Factors
To factor a number means to break it up into numbers that can be multiplied together to get the
original number. Example: 6 = 3 x 2 so, factors of 6 are 3 and 2 9 = 3 x 3 so, factors of 9 are 3 and 3.
Sometimes, numbers can be factored into different combinations.
A number can have many factors.
For example: What are the factors of 12?

● 3 × 4 = 12, so 3 and 4 are factors of 12

● 2 × 6 = 12, so 2 and 6 are also factors of 12
● and 1 × 12 = 12, so 1 and 12 are factors of 12 as well

So 1, 2, 3, 4, 6 and 12 are all factors of 12

And -1, -2, -3, -4, -6 and -12 also, because multiplying negatives makes a positive.

Example:
● 240 students in a group are to be seated in rows so that there are an equal number of
students in each row. Each of the following could be the number of rows EXCEPT.
A. 4 B. 20 C. 30 D. 40 E. 90

Answer: Option (E)

Explanation:
240/90 is not an integer. Because 90 is not a factor of 240. So, the answer is 90

Multiples
In math, the meaning of a multiple is the product result of one number multiplied by another number
(The result of multiplying a number by an integer, not by a fraction). Here, 56 is a multiple of 7.
For example -
● 12 is a multiple of 3, because 3 × 4 = 12
● −6 is a multiple of 3, because 3 × −2 = −6
● But 7 is NOT a multiple of 3

Example:
● 3 and 7 are factors of F. From this information, we can conclude that
A. 8 is a factor of F B. F is a multiple of 21 C. F=3 × 7
D. 21 is a multiple of 21 E. 3 and 7 are the only factors of F

Answer: Option (B)

Explanation:
Since, 21 is divisible by both 3 and 7, so F is a multiple of 21.

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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.

Example:
● What is the smallest number of apples that can distributed equally among 6, 9, 15, 18
students having a surplus of 3 apples each time?
A. 422 B. 362 C. 183 D. 62 e. none of these

Answer: Option (C)

Explanation:
H.C.F of 6, 9, 15 & 18= 90.
Option (A) 422 divided by 90 leaves a remainder of 62. So, it can be eliminated.
Option (B) 362 divided by 90 leaves a remainder of 2. So, it can be eliminated.
Option (C) 183 divided by 90 leaves a remainder of 3. So, it can be a possible option.
Option (D) 62 divided by 90 leaves a remainder of 62. So, it can be eliminated.
Only Option (C) matches the condition. So, Option (E) can be eliminated.
So, the answer is Option (C) 183.

Prime Number and Co-Primes

Prime Numbers
A prime number is a whole number greater than 1 whose only factors are 1 and itself. A factor is a
whole number that can be divided evenly into another number. The first few prime numbers are 2, 3,
5, 7, 11, 13, 17, 19, 23 and 29. Numbers that have more than two factors are called composite
numbers. The number 1 is neither prime nor composite.

Co-Primes
A number can never be called "a co-prime", rather, "co-primeness" is a relation that may or may not
hold between two numbers a and b. In other words, a and b may or may not be "co-prime to each
other".
What does it mean for a and b to be co-prime? That they share no common factors other than 1, or
equivalently, H.C.F. of and b is equal to 1.
For example, 2 and 5 are co-prime, because if d is a factor of 2 and d is also a factor of 5 (i.e., d is a
common factor of 2 and 5), then d has to be 1 (or −1, technically). In other words, H.C.F. of 2 and 5 is
equal to 1. However, 2 and 6 are not co-prime, because they share the common factor 2 and H.C.F.
of 2 and 6 is equal to 2.

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Class Practice

Type-1: Number Line Properties

1. If y is between 0 and 1, which of the following increases as y increases?

i. (1–y2) ii. (y–1) iii. 1-y

A. I and II B. II and III C. II only D. I, II, and III E. none of these

2. If x is between 0 and 1, which of the following increases as x increases?

2 2
I. 1–x II. 1-x III. 1/x
A. I and II B. II and III C. I & III only D. II only E. none of these

3. On a real number line, x1 = -4 and x2 = 14. What is the distance between these two points?
A. 4 B. -18 C. 18 D. 10 E. none of these

Type-2: Properties of Odd and Even Numbers

1. The sum of four consecutive odd positive integers is always

A. an odd number B. divisible by 4 C. divisible by 8

D. both (B) & (C). E. either (A) & (B)

2. If n is an odd integer, which of the following must be an even integer?

A. 13n–2 B. 3(2–n) C. (16n+24)/8 D. (6n + 12)/3 E. none of these

3. If x and y are positive even integers and z is an odd integer, which one of the following
statements cannot be true?

A. (x–z) is odd B. (x–z)2 y is even C. (x–z) y is even

D. (x–y)z is even E. none of these

Type-3: Divisibility
1. (xn – an) is completely divisible by (x - a), when
A. n is any natural number B. n is an even natural number
C. n is and odd natural number D. n is prime
E. n is a negative integer

2. The largest natural number which exactly divides the product of any four consecutive natural
numbers is:
A. 6 B. 12 C. 24 D. 120 E. Can’t be determined

3. If n is a natural number, then (6n2 + 6n) is always divisible by:

A. 6 only B. 6 and 12 both C. 12 only D. by 18 only E. None of these

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Type-4: Factors and Multiples
1. If x and y are two distinct positive integers divisible by 2, then which of the following is
necessarily divisible by 4? .
A. x + y B. x – y C. x2 + y2 D. 2x + y E. none of these

2. Find the value of Z if (x+1) is a factor of x3 + Zx + 3x2 – 2.

A. 6 B. 5 C. 4.5 D. 4 E. none of these

Type-5: L.C.M and H.C.F.

1. One third, one fifth, one-seventh and one-eighth of animals in a forest, which has fewer than
5000 animals, are all whole numbers and their sum is exactly number of fish in a river. What is
the number of fish in the river?
A. 3365 B. 4200 C. 3950 D. 4820 E. cannot be determined

2. What is the minimum number of apples that must be added to the existing stock of 770
chocolates so that the total stock can equally be divided among 8, 12 or 16 persons?
A. 32 B. 46 C. 58 D. 80 E. none of these

3. If x/2, x/3 and x/13 represent integers, then x can be

A. 42 B. 56 C. 70 D. 78 E. 126

Type-6: Prime numbers

1. How many integers between 100 and 110 are prime numbers?
A. 0 B. 1 C. 2 D. 3 E. 4

2. How many prime numbers are there between 65 and 100?

A. 6 B. 7 C. 8 D. 9 E. None of these

3. What is the closest prime number to 35?

A. 31 B. 33 C. 37 D. 39 E. 41

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