## Quantitative Aptitude Number System Tutorial (Study Material)

**Number System (Download Pdf) **

Numbers are collection of certain symbols or figures called digits. The most common Number System is Decimal System which has ten symbols, each representing a digit These are 0,1,2,3,4, 5,6,7,8 and 9.

See Also :

A combination of,these figures representing a number is called a numeral. We also have Binary Number System. It uses only 0 and 1. There are other Number Systems too.Every digit in a number has a face value and a place value. The face value of a digit equals the value of the digit itself, irrespective of its place in the numeral.

### Numbers and their Classification

As you could see in the last section, with its various number lines, there are a number of different ways to classify numbers. In fact, there are even more ways to classify numbers than last section displayed. This section will run through the most important and common classifications. You should memorize what each classification means.

Natural Numbers, Whole Numbers, Integers, and Rationals

**Natural Numbers**

The natural numbers, also called the counting numbers, are the numbers 1, 2, 3, 4, and so on. They are the positive numbers we use to count objects. Zero is not considered a “natural number.”

**Whole Numbers**

The whole numbers are the numbers 0, 1, 2, 3, 4, and so on (the natural numbers and zero). Negative numbers are not considered “whole numbers.” All natural numbers are whole numbers, but not all whole numbers are natural numbers since zero is a whole number but not a natural number.

**Integers**

The integers are …, -4 , -3 , -2 , -1 , 0, 1, 2, 3, 4, … — all the whole numbers and their opposites (the positive whole numbers, the negative whole numbers, and zero). Fractions and decimals are not integers. All whole numbers are integers (and all natural numbers are integers), but not all integers are whole numbers or natural numbers. For example, -5 is an integer but not a whole number or a natural number.

**Rational Numbers**

The rational numbers include all the integers, plus all fractions, or terminating decimals and repeating decimals. Every rational number can be written as a fraction a/b , where a and b are integers. For example, 3 can be written as 3/1, -0.175 can be written as -7/40 , and 1 1/6 can be written as 7/6. All natural numbers, whole numbers, and integers are rationals, but not all rational numbers are natural numbers, whole numbers, or integers.

We now have the following number classifications:

I. Natural Numbers

II. Whole Numbers

III. Integers

IV. Rationals

Numbers can fall into more than one classification. In fact, if a number falls into a category, it automatically falls into all the categories below that category. If a number is a whole number, for instance, it must also be an integer and a rational. If a number is an integer, it must also be a rational.

**Irrational Numbers**

There is a type of number that does not fall into any of our four categories. An irrational number is a number with a decimal that neither terminates or repeats. An irrational number cannot be written as a fraction a/b where a and b are integers. Plug in (the square root of 2) on a calculator and the screen will display a decimal that does not repeat itself, but that continues infinitely. This is because the square root of 2 is an irrational number.

There is no number which is both an irrational number and a natural number, whole number, integer, or rational number. If a number is irrational, it cannot fall into one of the four categories we previously outlined; and if a number falls into one of the four categories, it cannot be irrational.

**Real Numbers**

All the rational numbers and all the irrational numbers together form the real numbers. Every rational number is real, and every irrational number is real. For our purposes at this time, the real numbers constitute all the numbers. 0.45, 5/2, -0.726495 …, 18, and -65 1/4 are all real numbers.

Figure %: Classification of Numbers

If a number falls into a category, it also falls into all the categories below that category to which it is connected by a line. For example, -7 is an integer, so it is also a rational and a real number. The square root of 2 is an irrational number, so it is also a real number

### Test for Divisibility of Numbers

**Divisibility by 2 Rule**

Almost everyone is familiar with this rule which states that any even number can be divided by 2. Even numbers are multiples of 2. A number is even if ends in 0,2,4,6, or 8.

Examples of numbers that are even and therefore pass this divisibility test

2

0

4

-2

-312

31,102

Examples of numbers that are do not pass this divisibility test because they are not even.

3

-103

1.50

221

**Divisibility by 3 Rule**

Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.

Examples of numbers that are divisible by 3

12→1 + 2 = 3 And 3 is divisible by 3 so the number 12 is also divisible by 3.

36→3 + 6 = 9 And 9 is divisible by 3 which means that 36 is also.

102→1 + 0 + 2 = 3

100,002,000 = 1 +0 +0 +0 +0 +2 +0 + 0 +0 = 3 so this very large number passes this divisibility test.

36 = 3 + 6 = 9 and we all know that 9 ÷ 3 = 1 so this number satisfies the rule and is evenly divided by 3!

Examples of numbers that do not pass this test 14→

1+4 = 5 and since 5 is not divisible by 3, so 14 is also not.

124→1 + 2+ 4 = 7 which is no good, does not work.

100,002,001 = 1 +0 +0 +0 +0 +2 +0 + 0 + 1= 4 so this very large also does not pass this divisibility test.

**Divisibility by 4 Rule**

Rule: A number is divisible by 4 if the number’s last two digits are divisible by 4.

Use the divisibility calculator below to determine if any number is divisible by four. Type in any number that you want, and the calculator will use the rule for divisibility by 4 to explain the result.

Examples of numbers that are divisible by 4

112 → since the last two digits, 12, are divisible by 4, the number 112 is satisfies this rule and is also divisible by 4.

10,948 → the last two digits, 48, are divisible by 4. Therefore, the whole number is also.

100,002,088 = 88. Yep, this satisfies rule because 88 is divisible by 4!

-12,036 = 36 and 36 is evenly divided by 4, so -12,036 passes the test!

Examples of numbers that are do not pass this divisibility test

113 → since the last two digits, 13, are not divisible by 4, the whole number does not pass this divisibility test.

10,941 → the last two digits, 41, are not de visible by 4. Therefore, the whole number does not satisfy the rule for 4.

100,002,014 = 14 and 14 is no good, does not work.

-1,011 = 11 so 1,011 fails this test .

Ever wonder why these rules work. The test for 4 makes sense if you just break down the numbers. Think about what this rule says: “All that matters is whether or not the last two digits are divisible by 4.” Let’s look at why this rule is true.

Examine some three digit numbers

124 is the same as 100 + 24, and we know that 100 is divisible by 4 so all that matters here is whether or not 24, or the last two digits, are divisible by 4. The same could be said for any three digit number 224 = 200 + 24, and we know that 200 is divisible by 4 so again all that we’re worried about are these last two digits.

Any multiple of 100 is divisible by four! Whether you’re talking about 300, 700, 1000, 1100, 123,00– All of these multiples of 100 are divisible by 4, which means that all that we ever have to worry about is the last two digits!

**Divisibility by 5 Rule**

Rule: A number is divisible by 5 if the its last digit is a 0 or 5.

Examples of numbers that are divisible by 5 and satisfy this rule

10 → since the last digit is 0, 10 satisfies this rule and is divisible by 5

15 → since the last digit is 5, 15 satisfies this rule and is divisible by 5

45

-30

55

-105

12,340

Examples of numbers that fail this divisibility test.

17 → since the last digit is 7, 17 does not satisfy this rule and is not divisible by 5

118 → since the last digit is 8, 118 does not satisfy this rule and is not divisible by 5

-311 → Since the last digit is 1, 311 does not satisfy the rule for 5

-101

12,103

**Divisibility by 6 Rule**

Since 6 is a multiple of 2 and 3, the rules for divisibility by 6 are a combination of the rule for 2 and the rule for 3. In other words, a number passes this divisibility test only if it passes the test for 2 and test for 3.

Rule A number is divisible by 6 if it is even and if the sum of its digits is divisible by 3.

Examples of numbers that are divisible by 6

12 → satisfies both conditions:

1) 12 is even

2) the sum of its digits (1+2 =3) is divisible by 3. Therefore, 12 passes this test.

114 → satisfies both conditions

1) 1+1+4 = 6 which is divisible by 3

2) 114 is even

241,122 → This passes the test because it’s even and the sum of its digits can be evenly divided by 3.

Examples of numbers that are do not pass this divisibility test

207 → Fails the test since it’s not even. We don’t even have to see whether the second condition is satisfied since both conditions must be satisfied to pass this test. If only one of the two conditions (divisible by 2 and by 3) are not met, then the number does not satisfy the rule for 6.

241,124 → Although this number is even, the sum of its digits are not evenly divided by 6 so this fails the test.

**Divisibility by 8 Rule**

Rule A number passes the test for 8 if the last three digits form a number is divisible 8.

Examples of numbers that satisfy this rule and are divisible by 8

9,640 → 640 ÷ 8 = 80 so the whole number, 9,640, is divisible by 8

77, 184 → 184 ÷ 8 = 23 so 77,184 passes this divisibility test.

67, 536 → 536 is divisible by 8 ( 536 ÷ 8 = 67) so 67,536, is also.

-30 → 640 ÷ 8 = 80 so the whole number, 9640, passes this test.

20,233,322,496 → Well, maybe you were wondering if this divisibility rule was really helpful or not. Once you get a giant number like 20,233,322,496, you start to realize what a nice trick this is to have up your sleeve! All you have to do is divide 496 by 8 to learn that the entire number is divisible by 8.

– 316,145,664 → 664 passes this divisibility test.

Examples of numbers that are do not pass this divisibility test

9,801 → since 801 is not divisible by 8, 9,801 is not.

234,516 → Nope, no good. 516 is not evenly divided by 8 so the whole number fails the test!

-32,344,588 → 588 does not work, so -32,344,588 does not satisfy the rule for 8!

**Divisibility by 9 Rule**

Rule A number is divisible by 9 if the sum of the digits are evenly divisible 9.

Examples of numbers that satisfy this rule and are divisible by 9

4,518 → 4+5+1+8=18 and since 18 ÷9 = 2 , the whole number, 4,518, is divisible by 9

7,209 → 7+2+0+9 = 18, and by the same logic of the prior example, 7,209 passes this divisibility test.

6,993 → ,6993 is divisible by 9(6+9+9+3 = 27 & 27 ÷ 9 = 3) so 6,993 satisfies the rule for 9.

10,006,470 → Well, maybe you were wondering if this divisibility trick was really helpful or not. Once you get a giant number like 10,006,470, you start to realize what a nice trick this is to have up your sleeve! All you have to do is add the digits (1+ 6+4+7 = 18) to quickly see that the entire number is divisible by 9 (18÷9 = 2).

Examples of numbers that are do not pass this divisibility test

29 → 2+9 =11. Since 11 is not divisible by 9, 29 is not either.

6,992 → Nope, no good. 6+9+9+2 =26 which is not evenly divided by 9 so the whole number fails the test!

**Divisibility by 10 Rule**

Rule A number passes the test for 10 if its final digit is 0

Examples that pass this test

100

110

-110

1,320,320.

Examples of numbers that are do not pass this divisibility test

91,801 → last digit is not zero, so this does not work.

234,516 → Nope, this number does not satisfy the rule for 10.

-32,344,508 → Again, it all comes down to that last digit which just has to be zero!

**Divisibility by 11 Rule**

Rule A number passes the test for 11 if the difference of the sums of alternating digits is divisible by 11.(This abstract and confusing sounding rule is much clearer with a few examples)

Examples of numbers that satisfy this rule

946 → (9+6) – 4 = 11 which is, of course, evenly divided by 11 so 946 passes this divisibility test

10,813 → (1+8+3) – (0+1) = 12-1 =11. Yes, this satisfies the rule for 11!

25, 784 = → (2+ 7 + 4) – (5+8) = 13 – 13 =0 . Yes, this does indeed work. In case you found this last bit confusing, remember that any number evenly divides 0. Think about it, how many 11’s are there in 0? None, right. Well that means that 11 divides zero, zero times!

119,777,658 → (1+ 9 + 7 + 6 + 8) – (1+ 7 + 7 +5) = 31 – 20 = 11

examples of numbers that are do not pass this divisibility test

947 → (9+7) – 4 = 12 which is not divisible by 11

10,823 → (1+8+3) – (0+2) = 12- 2 =10. No, no good. This one fails!

35, 784 = → (3 + 7 + 4) – (5+8) = 14 – 13 = 1. No, does not satisfy the rule for 11!

12,347, 496, 132 = → (1+3+7+9+3) – (2 + 4 +4 + 6 + 3)= 23- 19 = 4

### General Properties of Divisibility

There are some general properties of div isibility that help in determining the divisibility of a natural number by other natural numbers (other than detailed in 1.2)

**Property 1**

If a number .v is divisible by another number y. then any number divisible by x, will also be divisible by y and by all the factors of y.

Example: The number 84 is divisible by 6. Thus any number that is divisible by 84. will also be divisible by 6 and also by the factors of 6. i.e. by 2 and by 3.

**Property 2**

If a number x is divisible by two or more than two co-prime numbers then x is also divisible by the product of those numbers.

Example: The number 2520 is divisible by 5. 4. 13 that are prime to each other (i.e. co-prime), so. 2520 will also be divisible by 20 (= 5 x 4). 65 (= 5 x 13). 52 (= 4 x 13).

**Property 3**

If two numbers ,r and y are divisible by a number p then their sum x+y is also divisible by p.

Example: The numbers 225 and 375 are both divisible by 5. Thus their sum = 225 + 375 = 600 will also be divisible by 5.

Note: It is also true for more than two numbers.

**Property 4**

If two numbers x and y are divisible by a number ‘p\ then their difference x – y is also divisible by p.

Example: The numbers 126 and 507 are both divisible by 3. Thus their difference = 507 – 126 = 381 will also be divisible by 3.

### Test of a Prime Number

A prime number is only divisible by l and by the number itself. The first prime number is 2. Every prime number other than 2 is odd, but every odd number is not necessarily a prime number. Again any even number (other than 2) cannot be a prime number. To test whether any given number (if odd) is a prime number or not. following steps are to be considered:

**Step 1** Find an integer (x) which is greater than the approximate square root of the given number.

**Step 2** Test the divisibility of the given number by every prime number less than x.

**Step 3** • If the given number is divisible by any of them in Step 2. then the given number is NOT a prime number.

- If the given number is not divisible by any of them in Step 2. then the given number IS a prime number.

**Example**: Consider a number 203. Test if it is a prime number or not.

**Step 1** The approximate square root of 203 is 14 plus. Take x= 15.

**Step 2** Check the divisibility of 203 by the prime numbers less than 15 i.e. by 2. 3, 5. 7, II, 13. Step 3 203 is divisible by 7. Thus, it is not a prime number.

### Division and Remainder

When a given number is not exactly divisible by any number, then there is a remainder number at the end of such division.

Suppose we divide 25 by 7 as.

7) 25 (3

21

—-

4

then, divisor = 7. dividend = 25

quotient = 3, and remainder = 4 So, we can represent it as

divisor )dividend ( quotient

———–

remainder

Thus dividend = (divisor x quotient ) + remainder

So. if a number x is divided by k. leaving remainder V and giving quotient q then the number can be found by using (i)

x = kq + r

Hence, if the number x is exactly divisible by k. then remainder = r = 0 .

x= kq

and so x/k =q implying that x is divisible exactly by k and q is an integer

### Remainder Rules

This rule is applied to find the remainder for the smaller divison, when the same number is divided by the two different divisors such that one divisor is a multiple of the other divisor and also the remainder for the greater divisor is known.If the remainder for the greater divisor = r, and the smaller divisor = d. then

**Rule-1** states, that when r >d. the required remainder for the smaller divisor will be the remainder found out by dividing the r by d

and when r <d . then the required remainder is r it self.

Example: If a number is divided by 527. the remainder is 42. What will be the remainder if it is divided by 17?

**Solution**: Here the same number is divided bv two divisors: 527 and 17.

Now. 527/17 =31. so, 527 is a multiple ol 17

Hence Rule 1 can be applied.

Remainder for the greater divisor (i.e., for 527) = 42 Smaller divisor = 17.

So. 17) 42(

34

————-

8

8 = required remainder for smaller divisor (i.e. 17)

Hence, if 527 is divided by 17, the remainder will be 8.

### Number Series

In the number series, some numbers arc arranged in a particular sequence. All the numbers form a series and change in a certain order. Sometimes, one or more numbers are wrongly put in the number series. One is required to observe the trend in which the numbers change in the series and Find out which number/ numbers mis Fit into the series. That number/numbers is the ODD NUMBER of the series.

Important Number Series

Following arc some of the important rules or order on which the number series can be made.

**1. Pure Series**

In this type of number series, the number itself obeys certain order so that the character of the series can be found out.

The number itself may be:

- perfect square
- perfect cube
- prime
- combination of above

**2. Difference Series**

Under this category, the change in order for the differences between each consecutive number of the series is found out as shown in Table l.l.

**3. Ratio Series**

Under this category, the change in order for the ratios between each consecutive number of the series is found out as shown in Table 1.2.

**4. Mixed Series**

Here, the numbers obeying various orders of two or more different type of series are arranged alternately in a single number series.

**5. Geometric Series**

Under this category, each successive number is obtained by multiplying (or dividing) the previous number with a fixed number. (See Table 1.2).

Example: 5, 35, 245, 1715, 12005_____

43923, 3993, 363. 33, 3____

**6. Two-tier Arithmetic Series**

Under this category, the differences of successive numbers form an arithmetic series. (See Table l.l)

**7. Twin Series**

Under this category, two series are alternatively placed in one.

### Three steps to Solve a Problem on Series

**Step 1** Determine whether the series is increasing, decreasing or alternating.

**Step 2** If the series is increasing or decreasing, then check:

- if change is slow or gradual, then it is a difference series.
- if the change is equally sharp, throughout, then it is a ratio series.
- if the rise is very sharp initially, but slows down later, then the series may be formed by adding squared, or cubed numbers.

If the series is alternating or irregular, there may be either a mix of two series or two different kinds of operations going on alternately.

**Step 3** Complete the series accordingly.

### Two line Number Series

A two-line number. series, as the name suggests, consists of number series in two lines. If one complete series is given in first line, with an incomplete series in second line, and it is given that the series in both the lines have the same definite rule, we need to work it out as follows:

Applying the very definite rule of the series in the first line, the series in second line can be completed. The pattern/type of series in the first line may be any of the types described in 1.7.

**Example**: 13 28 31 84 127 …

22 a b c d e . . .

### Sum Rules on Natural Numbers

### Base and Index

### Binary Number System

### Calculation in the Binary System

**Number System (Download Pdf) **