# NUMBERS - BASICS

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TYPES OF NUMBERS

Numbers can be broadly classified as – Real Numbers and Imaginary Numbers.

Real Numbers are those which can be represented on a number line.

Imaginary Numbers are square-root of negative numbers.

From the perspective of Quantitative Aptitude in Competitive Exams, Numbers will refer to only Real Numbers.

Real numbers can be classified into rational numbers and irrational numbers.

Rational Numbers:

Numbers which can be expressed in the form of P/Q, where Q≠ 0 and P and Q are integers are known as **‘Rational Numbers’**.

The above definition would include all the integers (positive, negative and zero), all the fractions and recurring decimal fractions.

Recurring Decimals are numbers which have one or more digits at the end of decimal number repeating continuosly .

Eg. 21.

It can be expressed as 21/1 , where value of P= 21 and Q=1. Hence it is a rational number.

Eg. 1.5666…

The decimal fraction is equivalent of 141/90. P= 141 and Q=90. Hence it is also a rational number.

Eg.0.33333…

The decimal fraction is equivalent of 1/3. P= 1 and Q=3. Hence it is also a rational number.

Irrational Numbers:

Numbers which do not fall under the category of rational numbers but can be represnted on a number line are all irrational.

Eg. √2,√7, 1.234589…..,0.46684429…

√2 = 1.41421356…..

The number cannot be expressed in P/Q form completely as the numbers after decimal point are neither terminating (ending) nor recurring (repeating)

π and e are also irrational numbers.

This may lead to a question that π is 22/7. Why is it still irrational?

The value of π is actually, 3.14159…

In order to make our calculations easier, we assume π to be 22/7

Integers:

They can be either negative, 0 or positive.

Eg. -3, -2, 0 ,10

Negative Integers: They fall to the left of 0 in the number line.

They start with -1 and go upto (-infinity)

Positive Integers: They fall to the right of 0 in the number line. They are also called ‘Natural Numbers’.

They start with 1 and go upto infinity.

Whole Numbers: Natural Numbers along with 0 are ‘whole numbers’

Even Numbers: The numbers which are divisible by 2 are known as ‘even numbers’

eg. 24 , 584, 962 etc.

Odd Numbers: The natural numbers which are not divisible by 2 are known as ‘odd numbers’.

eg. 35, 735 , 89 etc.

Prime Numbers: The numbers which are divisible by no numbers other than 1 and the number itself are ‘prime numbers’.

Eg. 2,3,5,7,11 etc.

Composite Numbers: The numbers greater than 1 which are divisible by numbers other than 1 and the number itself are composite numbers.

eg. 4,6,8,9,10 etc

1 is neither a prime nor a composite number.

2 is the only even prime number. All other even numbers are divisible by 2 and hence not prime.

Co-prime Numbers / Relatively Prime Numbers: A pair of numbers is said to be co-prime if no other number other than 1 divides both of them. ie. HCF of both numbers is 1.

eg. 4 and 21.

Number 4 is divisble by 1,2 and 4.

Number 21 is divisble by 1,3,7 and 21.

The only factor common between both the numbers is 1. Hence, co-prime.

Factorials: It is defined for positive numbers. It is multiplication of consecutive numbers from 1 to the given number. It is denoted generally using ‘!’ after the number.

Eg. 5! (read as 5 factorial)

5! = 5 x 4 x 3 x 2 x 1

=120

**Note: Both 0! and 1! equal to 1**

DIVISIBILITY PROPERTIES OF NUMBERS:

Divisibility by 2:

The last digit of the number has to be either 2,4,6,8 or 0.

Divisibility by 3:

The sum of the digits of number should be divisible by 3.

Eg. 813

Sum of digits = 8+1+3 =12.

12 is divisible by 3. hence, 813 is also divisible.

Divisibility by 4:

For divisibility by 2, we had considered only the last digit.

Similarly, for divisibility by 4, we consider last 2 digits. If the last 2 digits are divisible by 4, then the number is divisible by4.

Eg. 1804

The last 2 digits is 04. It is divisible by 4. hence the number is divisible by 4.

Divisiblity by 5:

The unit digit should be either ending in 5 or 0.

Divisibility by 6:

For a number to be divisible by any composite number, it should be divisible by its co-prime factors.

Co-prime numbers of 6 are 2 and 3.

Eg. 750.

The number should be divisible by both 2 and 3. It is divisible by 2 since, unit digit is 0. Sum of digits = 7+5+0 =12. So, it is divisible by 3 also.

Thus, 750 is divisible by 6.

Divisibility by 8:

For divisibility by 2, consider last digit

For divisibility by 4, consider last 2 digits

For divisibility by 8, consider last 3 digits

Eg.56488

Last 3 digits – 488 is divisible by 8. Thus the given number is divisible by 8.

Divisiblity by 9:

The sum of digits of the number should be divisible by 9.

Eg. 639

Sum of digits = 6+3+9 =18, whih is divisible by 9. Hence, the number 639 is divisible by 9.

Divisibility by 10:

The number should end in 0.

Divisibility by 11:

If the difference between the sum of digits at odd places and the sum of digits at even places is divisible by 11, then the number is either 0 or divisible by 11.

Eg. 14630

In the given number, Digits at odd places are : 1,6 and 0

Digits at even places are :4,3

Sum of digits at odd places =Sum of digits at even places = 7

Difference =0.

Hence the number is divisible by 11

FOR ALL OTHER COMPOSITE NUMBERS:

Divide the given number by co-prime factors (whose product is equal to divisor) of divisor and if the remainders are 0, then the number is divisible.

Eg. Dividing 5656 by 42.

The given number 5656 is to be divided by co-prime factors of 42.

We can consider either 7 and 6 or 3 and 14.

5656 is divisible by 7 but not by 6.

Similarly, 5656 is divisible by 14 but not by 3.

From both the ways, we arrive at the same result that 5656 is not divisble by 42.

FINDING PRIME FACTORS OF A NUMBER:

Any positive integer greater than 1 can be expressed as product of its prime factors.

24 = 4 x 6

=2 ^{2} x 2 x 3

=2 ^{3} x 3^{1}

Similarly, let us find out for 357

357 = 7 x 51

=3 ^{1} x 7 ^{1} x 17^{1}

Suppose, we had a prime number, say 43

43 = 43^{1}

PROPERTIES OF FACTORS OF A NUMBER:

Let a number A be a number which can be expressed as

A = K^{a } x L^{b } x M^{c }

(1) Number of factors of the number =

(a+1)x (b+1) x(c+1)

Eg. Let us find the number of factors of 96 by this method:

96 = 2 x 2 x 2 x 2 x 2 x 3

= 2 ^{5} x 3 ^{1}

Factors = (5+1) x (1+1)

= 6 x 2

= 12

Checking if this is correct by manually counting,

Factors of 96 are

1,2,3,4,6,8,12,16,24,32 and 96.

Count =12.

(2) Sum of factors of the number =

Eg.

Finding sum of factors of 96:

From above example,

96 = 2 ^{5}x 3 ^{1}

Addimg up the factors of 96, we get,

1+2+3+4+6+8+12+16+24+32+96

=252

Unique Property of Square Numbers:

Let us take 4 numbers and count the number of factors in them:

21,97,64 and 32

(i)21

21 = 3^{1} x 7 ^{1}

Number of factors = 2 x 2

= 4

(ii) 97

97 is a prime number. Thus,

97 = 97^{1}

Number of factors = 2

(iii)36

36 = 2^{2} x 3^{2}

Number of factors = 3 x 3

=9

(iv) 32

32 = 2^{5}

Number of factors = 6

Out of 4 cases, only in one case, the number of factors happened to be odd. ie. for 36

The number of factors of only square numbers will be odd.

4PROPERTIES OF DIVISION IN NUMBERS:

1. The number which has to be divided is known as ‘dividend’ and the number by which it is divided is known as ‘divisor’.

On dividing, we get a quotient – number of times divisor can completely divide dividend and a remainder.

Dividend = Quotient x Divisor + Remainder

Eg. Let us consider 45 to be divided by 7.

45 is the dividend and 7 is the divisor.

We get 6 as the quotient and 3 as the remainder.

45 = 6 x 7 +3

2. On subtracting the remainder from the dividend, the resulting number is completely divisible by divisor and the remainder is 0.

Eg. Considering the previous example,

On dividing 45 by 7, we get 3 as the remainder.

Subtract 3 from 45, we get 42 which is completely divisible by 7 and the remainder is 0.

3. The remainder has to be always less than the divisor. In any case, if remainder happens to be more than divisor, it means that it can be further simplified till the remainder is less than the divisor.

Eg. Let us take 3 numbers, 55, 18 and 73

As we can observe, 55 +18 =73

Remainder when 55 divided by 7 = 6

Remainder when 18 divided by 7 = 4

Remainder when 29 divided by 7 = 6+4 =10

It means that it is not the actual remainder and 10 can be divided by 7 again.

It will give a remainder of 3.

Checking, 73 when divided by 7, gives a remainder of 3.

4. If A+ B = C and all of them are divided by a common number, the remainders A_{R},B_{R},C_{R} form a relation:

A_{R} + B_{R} = C_{R}

Eg. Let us take 3 numbers in such a way that third number is a sum of 2 numbers and divide all of them by 7.

15, 14 and 29.

15 + 14 =29

Remainder when 15 divided by 7 = 1

Remainder when 14 divided by 7 = 0

Remainder when 29 divided by 7 = 0 + 1 = 1

5. If A- B = C and all of them are divided by a common number, the remainders A_{R},B_{R},C_{R} form a relation:

A_{R} – B_{R} = C_{R}

Eg. Let us take 3 numbers- 15, 114 and 129 and divide all of them by 7.

129 – 114 =15

Remainder when 129 divided by 7 = 3

Remainder when 114 divided by 7 = 2

Remainder when 15 divided by 7 = 1

6. If A x B = C and all of them are divided by a common number, the remainders A_{R},B_{R},C_{R} form a relation:

A_{R} x B_{R} = C_{R}

Eg. Let us take 3 numbers- 15, 12 and 180 and divide all of them by 7.

15 x 12 = 180

Remainder when 15 divided by 7 = 1

Remainder when 12 divided by 7 = 5

Remainder when 180 divided by 7 = 1 x 5

= 5

Let us take another pair of factors of 180 – 45 and 4, and try applying the same property.

180 = 45 x 4

Remainder when 45 divided by 7 = 3

Remainder when 4 divided by 7 = 4

Remainder when 180 divided by 7 = 3 x 4 =12

12 on division by 7 gives remainder 5.

CONCEPT OF NEGATIVE REMAINDER:

A remainder cannot be negative but for ease of calculations, we can assume remainders to be negative numbers.

When a number is divided by divisor D and gives remainder R, the remainder can also be expressed as (R-D) for ease.

Eg.

Find remainder of 27 x 27 x 27 x 27 when divided by 14.

Soln:.

On dividing 27 by 14, we get a remainder of 13. Using remainder as 13 will be cumbersome in calculations.

Remainder = 13 x 13 x 13 x 13

Since, the remainder is huge, we will have to divide continuously by 14 till the remainder is less than 14.

Hence, we assume the remainder as

(13 -14)

= -1

Now solving the same problem,

Remainder = (-1) x (-1) x (-1) x (-1)

= 1

SUCCESSIVE DIVISION:

Let us take 2 numbers – 35 and 3 and divide 35 by 3.

We get a quotient of 11 and remainder of 2.

Th**e** quotient is large enough to be again divided by 3.

On dividing 11 by 3, we get,

Quotient of 3 and remainder of 2

On again dividing quotient by divisor, we get quotient of 1 and remainder of 0.

On again dividing quotient by divisor, we get quotient of 0 and remainder of 1.

The same can be expressed as :

On successive division we get the remainders – 2,2,2 and 1 respectively.

Note: The order of remainders is very important. Getting remainders 2,2,2 and 1 is not same as 2,1,2 and 2.