Quantitative Aptitude - SPLessons

Integers

Chapter 55

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Integers

shape Introduction

The numbers (1, 2, 3, … ), along with their negatives, (−1, −2, −3, …) and zero, 0 constitute the integer set.

Therefore, { … , −3, −2, −1, 0, 1, 2, 3, … } represents the set of integers.

The positive integers are greater than 0, the negative integers are less than 0, and 0 is neither positive nor negative.

An integer is always obtained whenever integers are added, subtracted, or multiplied.

Example: 5, 1, 5, 8, 97, and 3,043.


shape Methods

Number line is formed when a numerical value is assigned to each point on a line.


For example the above line represents a number line.

Point A is located at negative one likewise point B is located at positive one.


Number lines are useful to strive the location of the points whereas the Origin which is usually assigned the value of Zero. Also known as this line is broken into two regions perhaps the half in the right contains positive numbers and the half in the left contains negative numbers.


Positive integers: 1, 2, 3, 4, 5 …….

Negative Integers: -1, -2, -3, -4, -5 …….

Zero: 0


In general,

Integers are whole numbers such as positive integers like 1, 2, 3, 4, 5 ……. together with negative integers -1, -2, -3, -4, -5 ……. along with Zero form the set of integers in order to represent by segmentation as follows


Set of Integers: {………, -3, -2, -1, 0, 1, 2, 3, ………}


Enclose the numbers by using flower brackets.

Ellipses mean that the numbers keep going forever and ever with the implied pattern.


In other words, the numbers keep going in both the directions –infinity to the left and +infinity to the right also known as the set of integers are listed consecutively means that the integers are listed in order of increasing value without any integers missing in between them.


On the number line moving away from the zero, positive integers are greater than zero and get bigger.

For example: 2 > 1

On the number line moving away from the zero, negative integers are less than zero and get smaller.

For example: -2 < -1

Zero is considered as an integer since it is neither a positive nor negative number. Zero doesn’t consider any particular sign. It is considered as sign “neutral”.


Any number between integers like \(\frac{1}{2}\), \(\frac{1}{3}\) and –\(\frac{1}{4}\) are not considered as integers. They are known as fractions.

Remember the set of integers do not contain fractional parts of a number. Moreover, fractions and decimals cannot be listed consecutively like integers can.


Operations of Integers:

Different type of operations that can be used for integers is discussed below:

The four basic operations that can be applied to integers are

    1. Addition

    2. Subtraction

    3. Multiplication

    4. Division


Before discussing the operations, let’s talk about the absolute value.


The absolute value of a number is equal to its distance away from 0 on the number line, which means that the absolute value of any number is always positive whether the number itself is positive or negative.


The symbol for absolute value is a set of double lines as shown below

\(\mid \mid\) → Absolute Value symbol


The positive number of any pair of opposite non-zero numbers is called the absolute value of each number in the pair.

For Example, 1 and -1 form positive pairs. One is a positive integer and the other is a negative integer.

The absolute value of +1 = 1

The absolute value of -1 = 1

In general,

If ‘a’ is positive integer then the absolute value of a = a

\(\mid a \mid\) = a

If ‘a’ is negative integer then the absolute value of -a = a

\(\mid -a \mid\) = a

If ‘a’ is zero then the absolute value of a = 0

\(\mid 0 \mid\) = 0


Addition and Subtraction:

Say ‘a’ and ‘b’ are two integers, then

Case 1: If ‘a’ and ‘b’ are both positive integers then add their absolute values and prefix the positive sign.

Example: 3 + 7 = 10


Case 2: If ‘a’ and ‘b’ are both negative then add their absolute values and prefix the negative sign.

Example: – 6 + ( – 2) = – (6 + 2) = – 8


Case 3: If ‘a’ is positive and ‘b’ is negative and ‘a’ has a greater absolute value than ‘b’ subtract ‘b’ from ‘a’ and prefix a positive sign to the result.

Example: 8 + (- 5) = 8 – 5 = 3


Case 4: If ‘a’ is positive and ‘b’ is negative and ‘a’ has a greater absolute value than ‘a’ subtract ‘a’ from ‘b’ and prefix a positive sign to the result.

Example: 4 + (-9) = – (9 – 4) = – 5


Case 5: If ‘a’ and ‘b’ are opposites then their sum is equal to zero.

Example: 2 + (- 2) = 0



Multiplication:

If two numbers have the same sign, their product is positive.

(+) x (+) = +

Example: (5) x (6) = 30

(-) x (-) = +

Example: (-5) x (-6) = 30


If two numbers have opposite signs, then their product is negative

(-) x (+) = –

Example: (-5) x (6) = -30

(5) x (-6) = 30


Product of the integers:

Factor (Divisor)

When integers are multiplied, each of the multiplied integers is called a factor also known as a divisor of the resulting product.

Example: (6) (5) = 30

Here 6 and 5 are the factors or divisors of 30

Example: (2)(3)(10) = 60

Here 2, 3, 10 are the factors or divisors of 60.

But notice that

(4)(15) = 60 and (5)(12) = 60

So 4, 5, 12, 15 are the factors or divisors of 60

Hence, positive factors (divisors) of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Remember that the negatives of these integers are also the factors of 60 i.e.

-1, -2, -3, -4, -5, -6, -10, -12, -15, -20, -30, -60

Since negative two times negative 30 is equal to 60

Negative four times negative 15 is also equal to 60


Multiples

A multiple of an integer is that number multiplied by an integer.

Example: (10) (-2) = -20


Example: Find the factors of 100

100 = 1 x 100

= 2 x 50

= 4 x 25

= 5 x 20

= 10 x 10

So the factors of 100 = 1, 2, 4, 5, 10, 20, 25, 50, 100 and their opposites i.e.

-1, -2, -4, -5, -10, -20, -25, -50, -100

Hence, 100 has the multiples of 18 integers i.e. 9 positive and 9 negative.


Least Common Multiple (LCM):

If ‘a’ and ‘b’ are two non-zero integers,

The least common multiple (LCM) of these two integers is the least positive integer that is a multiple of both ‘a’ and ‘b’.

Example: LCM of 30, 75

Multiples of 30: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, ….

Multiples of 75: 75, 150, 225, 300, 375, 450, …….

Common multiples: 150, 300, 450, ….

Therefore, the least of these is 150

Hence LCM (30, 75) = 150


Greatest Common Factor (GCF):

The greatest common divisor (GCD) also known as the greatest common factor (GCF) of two non zero integers ‘a’ and ‘b’ is the greatest positive integer that is a divisor also known as a factor of both ‘a’ and ‘b’.

Example: GCF of 30, 75

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Factors of 75: 1, 3, 5, 15, 25, 75

Common factors: 1, 3, 5, 15

Greatest of these is 15.

So GCF (30, 75) is 15.


Division:

When an integer ‘a’ is taken and divided by an integer ‘b’, where ‘b’ is a divisor or factor of ‘a’, the result is always a divisor of ‘a’.

\(\frac{a}{b}\) = q

Example: \(\frac{60}{6}\) = 10

Here, ‘b’ is not a divisor of ‘a’


Quotient and remainder

In general,

When a positive integer say ‘a’ is divided by positive integer ‘b’

\(\frac{a}{b}\)

First find the Greatest multiple of ‘b’ that is less than or equal to ‘a’.

This particular multiple can be expressed as qb,

a = qb

Where ‘q’ is the quotient

qb ≤ a

The remainder is equal to ‘a’ minus that multiple of ‘b’.

r = a – qb

Remember that remainder is always greater than or equal to zero and less than ‘b’.

r ≥ 0 and r < b


Examples:

Case 1: Quotient and remainder

\(\frac{100}{45}\) = 2 remainder 10

a = qb + r

100 = (2)(45) + 10

Remainder = 10

Quotient = 2


Case 2: Quotient only

\(\frac{24}{4}\) = 6 remainder 0

24 = (6) (4) + 0

Remainder = 0

Quotient = 6


Case 3: Remainder only

\(\frac{6}{24}\) = 0 remainder 6

6 = (0) (24)

Quotient = 0

Remainder = 6


Divisibility of Integers

Divisibility rules


  • An integer is divisible by 2 if its units digit is divisible by two.
    Example: 1,562,158 is divisible by 2 because the units digit 8 is divisible by 2.

  • An integer is divisible by 3 if the sum of its digits is divisible by three.
    Example: 5616
    5 + 6 + 1 + 6 = 18 which is divisible by 3.

  • An integer is divisible by 4 if its last two digits form a number that’s divisible by four.
    Example: 712 is divisible by 4 since the last two digits are divisible by 4.

  • An integer is divisible by 5 if its units digit is either 0 or 5.
    Example: 105 is divisible by 5 since its last digit is 5.

  • An integer is divisible by 6 if it is divisible by both 2 and 3.
    Example: 204 is divisible by 6 since it is divisible by both 2 and 3.

  • An integer is divisible by 9 if the sum of its digits is divisible by nine.
    Example: 61,326 is divisible by 9

  • An integer is divisible by 10 if its units digit is zero.
    Example: 100


Special Integers:

Even and Odd Integers:

If an integer is divisible by 2 evenly, in other words obtain a remainder of 0 then the integer is called an even integer. Otherwise it is an odd integer.

A characteristic of positive Odd integers is that when divide them by 2, the remainder is always 1.
The set of even integers is denoted as follows:

Even integers = {……, -6, -4, -2, 0, 2, 4, 6 …….}

The set of odd integers is denoted as follows:

Odd integers = {….., -5, -3, -1, 1, 3, 5 ………}


Sum of Even and Odd numbers

Even + Even = Even

2 + 4 = 6

Odd + odd = Even

3 + 3 =6

Even + Odd = Odd

2 + 3 = 5


Product of Even and Odd Numbers

Even x Even = Even

2 x 4 = 8

Odd x Odd = Odd

3 x 5 = 15

Even x Odd = Even

4 x 3 = 12


Prime numbers

A prime number is an integer greater than 1 that has only two positive divisors 1 and itself.

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, …….

Example: Check if 10 is a prime number?

10 = 1 x 10

= 2 x 5

10 is not a prime number since it has more than 2 divisors.

shape Samples

1. Quantitative Comparison

Quantity A Quantity B
The least prime number greater than 24 The greatest prime number less than 28
    A. Quantity A is greater.
    B. Quantity B is greater.
    C. The two quantities are equal.
    D. The relationship cannot be determined from the information given.


Solution:

    For the integers greater than 24, note that 25, 26, 27, and 28 are not prime numbers, but 29 is a prime number, as are 31 and many other greater integers. Thus, 29 is the least prime number greater than 24, and Quantity A is 29. For the integers less than 28, note that 27, 26, 25, and 24 are not prime numbers, but 23 is a prime number, as are 19 and several other lesser integers. Thus, 23 is the greatest prime number less than 28, and Quantity B is 23. Thus, the correct answer is Choice A, Quantity A is greater.


2. Which of the following integers are multiples of both 2 and 3? Indicate all such integers.

    A. 8
    B. 9
    C. 12
    D. 18
    E. 21
    F. 36


Solution:

    First identify the multiples of 2, which are 8, 12, 18 and 36, and then among the multiples of 2 identify the multiples of 3, which are 12, 18 and 36.

    Alternatively, if you realize that every number that is a multiple of 2 and 3 is also a multiple of 6, you can identify the choices that are multiples of 6. The correct answer consists of Choices C (12), D (18) and F (36).


3. What should least number be subtracted from 2590 so that the remainder when divided by 9, 11 and 13 will leave in each case the same remainder 6? Which of the following statements is true? Indicate the correct option.

    A. 2
    B. 3
    C. 6
    D. 10
    E. 14


Solution:

    The remainder will be divisible by the LCM of 9, 11 and 13

    LCM = 1287

    1287*2=2574

    2574+6=2580

    2590-2580=10

    Hence, the required number is 10


4. Quantitative Comparison

Column A Column B
Number of factors of 20 that are smaller than 40 Number of multiples of 20 that are smaller than 45
    A. Quantity A is greater.
    B. Quantity B is greater.
    C. The two quantities are equal.
    D. The relationship cannot be determined from the information given.


Solution:

    Factors of 20 = 1, 2, 4, 5, 10, 20

    Multiples of 20 that are smaller than 45 = 20, 40

    So, option A is correct.


5. n and p are integers greater than 1
5n is the square of a number
75np is the cube of a number.
The smallest value for n + p is

    A. 14
    B. 18
    C. 20
    D. 30
    E. 50


Solution:

    The smallest value for n such that 5n is a square is 5.

    75np can now be written as 75 x 5 x p.

    This gives prime factors…. 3 x 5 x 5 x 5 x p

    To make the expression a perfect cube, p will have to have factors 3 x 3, and hence p =9

    n + p = 5 + 9 = 14