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SSC CGL Exam  >  SSC CGL Notes  >  Famous Books Course for Competitive Exams  >  RS Aggarwal Summary: Number system

RS Aggarwal Summary: Number system

Introduction

The Number System forms the backbone of quantitative aptitude in competitive examinations. This chapter is crucial as questions from this topic appear consistently across all major competitive exams, often accounting for 15-20% of the quantitative section. Mastering number systems, divisibility rules, and related concepts not only helps solve direct questions but also builds a strong foundation for topics like HCF-LCM, averages, percentages, and algebra. A thorough understanding of this chapter can significantly boost your overall score and problem-solving speed.

1. Numbers and the Hindu-Arabic System

The Hindu-Arabic number system uses ten digits: $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$

A numeral is a group of digits representing a number, written using a place-value chart:

Ten CroresCroresTen LakhsLakhsTen ThousandsThousandsHundredsTensOnes

Writing Numbers in Words and Figures

Example: The numeral $4,38,09,67$ is written as "Forty-three lakh eighty thousand nine hundred sixty-seven"

"Nine crore four lakh six thousand two" is written as $9,04,06,002$

2. Face Value and Place Value

Face Value

The face value of a digit is its own value, regardless of its position in the numeral.

Example: In $6872$, the face value of $2$ is $2$, of $7$ is $7$, of $8$ is $8$, and of $6$ is $6$.

Place Value (Local Value)

The place value depends on the position of the digit:

  • Place value of ones digit = (digit) $\times 1$
  • Place value of tens digit = (digit) $\times 10$
  • Place value of hundreds digit = (digit) $\times 100$, and so on
Example: In $70984$
  • Place value of $4 = 4 \times 1 = 4$
  • Place value of $8 = 8 \times 10 = 80$
  • Place value of $9 = 9 \times 100 = 900$
  • Place value of $7 = 7 \times 10000 = 70000$
Note: Place value of $0$ is always $0$, regardless of position.

3. Types of Numbers

3.1 Natural Numbers

Counting numbers: $\{1, 2, 3, 4, 5, \ldots\}$

3.2 Whole Numbers

Natural numbers including zero: $\{0, 1, 2, 3, 4, \ldots\}$

3.3 Integers

All counting numbers, zero, and their negatives: $\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$

  • Positive integers: $\{1, 2, 3, 4, \ldots\}$
  • Negative integers: $\{-1, -2, -3, -4, \ldots\}$
  • Non-negative integers: $\{0, 1, 2, 3, 4, \ldots\}$

3.4 Even Numbers

Numbers divisible by $2$: $\{0, 2, 4, 6, 8, 10, \ldots\}$

3.5 Odd Numbers

Numbers not divisible by $2$: $\{1, 3, 5, 7, 9, 11, \ldots\}$

3.6 Prime Numbers

Numbers with exactly two factors ($1$ and itself).

All primes less than 100: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97$

Example: Test if $137$ is prime

Since $12^2 > 137$, test divisibility by primes less than $12$: $2, 3, 5, 7, 11$.

None divides $137$, so $137$ is prime.

Example: Test if $319$ is prime

Since $18^2 > 319$, test divisibility by primes less than $18$: $2, 3, 5, 7, 11, 13, 17$.

$11$ divides $319$, so $319$ is not prime.

3.7 Composite Numbers

Numbers with more than 2 factors. The least composite number is $4$.

3.8 Perfect Numbers

A number equal to the sum of its factors (excluding itself).

Examples: $6$ and $28$
  • Factors of $6$: $1, 2, 3, 6 \implies 1 + 2 + 3 = 6$
  • Factors of $28$: $1, 2, 4, 7, 14, 28 \implies 1 + 2 + 4 + 7 + 14 = 28$

3.9 Co-primes (Relative Primes)

Two numbers whose HCF is $1$.

Examples: $(2, 3)$, $(8, 9)$

3.10 Twin Primes

Two prime numbers whose difference is $2$.

Examples: $(3, 5)$, $(5, 7)$, $(11, 13)$

3.11 Rational Numbers

Numbers expressible as $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.

Examples: $\frac{1}{8}, \frac{2}{11}, -\frac{8}{3}, 0, 6, 5$

3.12 Irrational Numbers

Numbers with non-terminating, non-repeating decimal expansions.

Examples: $\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}, \pi, e, 0.231764735\ldots$

4. Important Facts About Numbers

  • All natural numbers are whole numbers
  • Not all whole numbers are natural numbers ($0$ is not natural)

Addition/Subtraction Rules:

  • Even $\pm$ Even = Even
  • Odd $\pm$ Odd = Even
  • Even $\pm$ Odd = Odd

Multiplication Rules:

  • Even $\times$ Even = Even
  • Odd $\times$ Odd = Odd
  • Even $\times$ Odd = Even

Important Points:

  • The smallest prime number is $2$
  • The only even prime number is $2$
  • The first odd prime number is $3$
  • $1$ is neither prime nor composite
  • The least composite number is $4$
  • The least odd composite number is $9$

5. Important Algebraic Formulae

$(a + b)^2 = a^2 + b^2 + 2ab$
$(a - b)^2 = a^2 + b^2 - 2ab$
$(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)$
$(a + b)^2 - (a - b)^2 = 4ab$
$(a + b)^3 = a^3 + b^3 + 3ab(a + b)$
$(a - b)^3 = a^3 - b^3 - 3ab(a - b)$
$a^2 - b^2 = (a + b)(a - b)$
$(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$
$a^3 + b^3 = (a + b)(a^2 + b^2 - ab)$
$a^3 - b^3 = (a - b)(a^2 + b^2 + ab)$
$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$
Example: Simplify $796 \times 796 - 204 \times 204$

Using $a^2 - b^2 = (a + b)(a - b)$:

$$= (796 + 204)(796 - 204) = 1000 \times 592 = 592000$$

Example: Find the square root of $4a^2 + b^2 + c^2 + 4ab - 2bc - 4ac$

$$= (2a)^2 + b^2 + (-c)^2 + 2(2a)(b) + 2(b)(-c) + 2(2a)(-c)$$

$$= (2a + b - c)^2$$

$\therefore$ Square root = $2a + b - c$

6. Tests of Divisibility

6.1 Divisibility by 2

A number is divisible by $2$ if its unit digit is $0, 2, 4, 6$, or $8$.

Example:

$58694$ is divisible by $2$; $86945$ is not.

6.2 Divisibility by 3

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

Example:

In $695421$, sum $= 6 + 9 + 5 + 4 + 2 + 1 = 27$, divisible by $3$

$\therefore$ $695421$ is divisible by $3$

6.3 Divisibility by 9

A number is divisible by $9$ if the sum of its digits is divisible by $9$.

Example:

In $246591$, sum $ = 27$, divisible by $9$

$\therefore$ $246591$ is divisible by $9$

Example: Which digit should replace $*$ in $197*5462$ to make it divisible by $9$?

Sum $= 1 + 9 + 7 + x + 5 + 4 + 6 + 2 = 34 + x$

For divisibility by $9$, $x = 2$ (making sum $= 36$)

6.4 Divisibility by 4

A number is divisible by $4$ if its last two digits form a number divisible by $4$.

Example:

$6879376$ is divisible by $4$ ($76$ is divisible by $4$)

6.5 Divisibility by 8

A number is divisible by $8$ if its last three digits form a number divisible by $8$.

Example:

$16789352$ is divisible by $8$ ($352$ is divisible by $8$)

6.6 Divisibility by 10

A number is divisible by $10$ if its unit digit is $0$.

6.7 Divisibility by 5

A number is divisible by $5$ if its unit digit is $0$ or $5$.

6.8 Divisibility by 11

A number is divisible by $11$ if the difference between the sum of digits at odd places and the sum at even places is $0$ or divisible by $11$.

Example: Show that $4832718$ is divisible by $11$

Odd places sum: $8+7+3+4 = 22$

Even places sum: $1+2+8 = 11$

Difference: $22-11 = 11$ (divisible by $11$)

$\therefore$ $4832718$ is divisible by $11$

6.9 Divisibility by 25

A number is divisible by $25$ if its last two digits form $00$ or a number divisible by $25$.

Example:

$63875$ is divisible by $25$ ($75$ is divisible by $25$)

6.10 Divisibility by 7 or 13

Divide the number into groups of $3$ digits from right. Find the difference between sum of groups at odd and even positions. If this is $0$ or divisible by $7/13$, the number is divisible by $7/13$.

Example: $4537792 \to 4|537|792$

$(792 + 4) - 537 = 259$, divisible by $7$

$\therefore$ $4537792$ is divisible by $7$

6.11 Divisibility by 16

A number is divisible by $16$ if its last four digits form a number divisible by $16$.

6.12 Composite Divisibility Tests

  • By 6: Divisible by both $2$ and $3$
  • By 12: Divisible by both $3$ and $4$
  • By 15: Divisible by both $3$ and $5$
  • By 18: Divisible by both $2$ and $9$
  • By 14: Divisible by both $2$ and $7$
  • By 24: Divisible by both $3$ and $8$
  • By 40: Divisible by both $5$ and $8$
  • By 80: Divisible by both $5$ and $16$
Important Note: If a number is divisible by $p$ and $q$ (co-primes), then it's divisible by $pq$. If $p$ and $q$ are not co-primes, this may not hold.
Example: Is $52563744$ divisible by $24$?

$24 = 3 \times 8$ (co-primes)

Sum of digits $= 36$ (divisible by $3$) ✓

Last $3$ digits $= 744$ (divisible by $8$) ✓

$\therefore$ $52563744$ is divisible by $24$

7. Multiplication Shortcuts

7.1 Distributive Law

$a \times (b + c) = a \times b + a \times c$
$a \times (b - c) = a \times b - a \times c$
Example: $567958 \times 99999$

$$= 567958 \times (100000 - 1)$$

$$= 56795800000 - 567958$$

$$= 56795232042$$

7.2 Multiplication by $5^n$

Put $n$ zeros to the right and divide by $2^n$.

Example: $975436 \times 625 = 975436 \times 5^4$

$$= \frac{9754360000}{16} = 609647500$$

7.3 Squaring Numbers

Example: $1605 \times 1605 = (1600 + 5)^2$

$$= 1600^2 + 5^2 + 2(1600)(5)$$

$$= 2560000 + 25 + 16000 = 2576025$$

8. Factorial of a Number

$$n! = n \times (n-1) \times (n-2) \times \cdots \times 3 \times 2 \times 1$$

Example: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
Note: $0! = 1$

9. Modulus of a Number

$$|x| = \begin{cases} x, & \text{when } x \geq 0 \\ -x, & \text{when } x < 0 \end{cases}$$

Examples: $|-5| = 5$, $|4| = 4$, $|-1| = 1$

10. Greatest Integral Value

$[x]$ denotes the greatest integer not exceeding $x$.

Examples: $[1.35] = 1$, $\left[\frac{11}{4}\right] = [2.75] = 2$

11. Division Algorithm (Euclidean Algorithm)

$$\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}$$
Example: On dividing $15968$ by a certain number, quotient is $89$ and remainder is $37$. Find the divisor.

$$\text{Divisor} = \frac{\text{Dividend} - \text{Remainder}}{\text{Quotient}}$$

$$= \frac{15968 - 37}{89} = 179$$

Example: A number when divided by $114$ leaves remainder $21$. If divided by $19$, find the remainder.

Number $= 114k + 21 = 19(6k) + 19 + 2 = 19(6k+1) + 2$

$\therefore$ Remainder when divided by $19 = 2$

12. Important Divisibility Properties

  • $(x^n - a^n)$ is divisible by $(x - a)$ for all values of $n$
  • $(x^n - a^n)$ is divisible by $(x + a)$ for all even values of $n$
  • $(x^n + a^n)$ is divisible by $(x + a)$ for all odd values of $n$
Example: Find remainder when $9^6 + 7$ is divided by $8$

$(9^6 - 1)$ is divisible by $(9 - 1) = 8$

$\therefore$ $(9^6 - 1) + 8$ is divisible by $8$

$\therefore$ $(9^6 + 7)$ gives remainder $0$

Example: Find remainder when $(397)^{3589} + 5$ is divided by $398$

$(397^{3589} + 1)$ is divisible by $(397 + 1) = 398$ [odd power]

$\therefore$ $(397^{3589} + 1) + 4$ gives remainder $4$

$\therefore$ Remainder $= 4$

13. Counting Zeros in Factorials

To find zeros at the end of $N = 1 \times 2 \times 3 \times \cdots \times 100$:

Find highest power of $5$ (limiting factor with $2$)

Example:

Highest power of $5 = \left[\frac{100}{5}\right] + \left[\frac{100}{25}\right] = 20 + 4 = 24$ zeros

14. Unit Digits in Products

Example: Find unit digit in $(2467)^{153} \times (341)^{72}$

Unit digit = Unit digit of $7^{153} \times 1^{72}$

$7^4$ gives unit digit $1$

$\therefore$ $7^{152}$ gives unit digit $1$

$\therefore$ $7^{153}$ gives unit digit $7$

$1^{72}$ gives unit digit $1$

$\therefore$ Unit digit $= 7 \times 1 = 7$

15. Successive Division

Example: A number successively divided by $3$, $5$, and $8$ leaves remainders $1$, $4$, and $7$. Find the number.

Working backwards:

$z = 8(1) + 7 = 15$

$y = 5(15) + 4 = 79$

$x = 3(79) + 1 = 238$

Practice Examples

Example 1: Basic Operations - Simplify: $8888 + 888 + 88 + 8$

$$\begin{align} & 8888 \\ & 888 \\ & 88 \\ + & 8 \\ \hline & 9872 \end{align}$$

Answer: $9872$

Example 2: Using Formulae - Simplify: $(387^2 + 113^2 + 2 \times 387 \times 113)$

This is of form $a^2 + b^2 + 2ab = (a+b)^2$

$$= (387 + 113)^2$$

$$= 500^2$$

$$= 250000$$

Example 3: Cube Identities - Simplify: $\frac{789^3 + 211^3}{789^2 - 789 \times 211 + 211^2}$

Using $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$

$$= \frac{(789 + 211)(789^2 - 789 \times 211 + 211^2)}{789^2 - 789 \times 211 + 211^2}$$

$$= 789 + 211$$

$$= 1000$$

Example 4: Testing Prime Numbers - Test if $571$ is prime

$24^2 > 571$

Prime numbers $< 24$: $2, 3, 5, 7, 11, 13, 17, 19, 23$

$571$ is not divisible by any of these

$\therefore$ $571$ is prime

Example 5: Divisibility - Is $57463822$ divisible by $11$?

Odd places: $2 + 8 + 6 + 7 = 23$

Even places: $2 + 3 + 4 + 5 = 14$

Difference: $23 - 14 = 9$ (not divisible by $11$)

$\therefore$ Not divisible by $11$

Example 6: Finding Missing Digits - What values of $M$ and $N$ make $M39048458N$ divisible by both $8$ and $11$?

For divisibility by $8$: Last $3$ digits $58N$ must be divisible by $8$

$\therefore$ $N = 4$ ($584$ is divisible by $8$)

For divisibility by $11$:

$(8 + 4 + 4 + 9 + M) - (4+5+8+0+3) = 0$ or multiple of $11$

$(25 + M) - 20 = M+5$

For divisibility: $M = 6$

$\therefore$ $M = 6, N = 4$

Example 7: Remainder Problems - When divided by $6$, a number leaves remainder $3$. What remainder does its square give when divided by $6$?

Number $= 6k + 3$

Square $= (6k + 3)^2 = 36k^2 + 36k + 9$

$= 36k^2 + 36k + 6 + 3$

$= 6(6k^2 + 6k + 1) + 3$

$\therefore$ Remainder $= 3$

Example 8: Smallest/Largest Numbers - Find the smallest $5$-digit number exactly divisible by $476$

Smallest $5$-digit number $= 10000$

$10000 \div 476$ gives remainder $4$

Number to add $= 476 - 4 = 472$

$\therefore$ Required number $= 10472$

Example 9: Zeros in Products - Find number of zeros at end of $1 \times 2 \times 3 \times \cdots \times 100$

Highest power of $5$ in $100!$

$$= \left[\frac{100}{5}\right] + \left[\frac{100}{25}\right]$$

$$= 20 + 4$$

$$= 24 \text{ zeros}$$

Example 10: 99-Digit Number Divisibility - A $99$-digit number formed by writing $1234567891011\ldots5859$. Find remainder when divided by $16$.

Remainder = Remainder when last $4$ digits $(5859)$ divided by $16$

$5859 \div 16 = 366$ remainder $3$

$\therefore$ Remainder $= 3$

Key Takeaways

  • Master divisibility rules - they save significant time
  • Learn to recognize patterns in prime numbers
  • Practice identifying which formula applies to each problem
  • Understand the division algorithm thoroughly
  • Work systematically through successive division problems
  • Remember special properties for powers and remainders
  • Use shortcuts for multiplication wherever possible
  • Practice unit digit calculations for speed
  • Understand factorial properties for counting zeros
  • Apply co-prime concepts for composite divisibility
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FAQs on RS Aggarwal Summary: Number system

1. What is the Hindu-Arabic numeral system?
Ans. The Hindu-Arabic numeral system is the most widely used numeral system in the world today. It is a base-10 system that originated in India and was later transmitted to the Islamic world and then to Europe. It consists of ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, and employs place value to denote the value of a digit based on its position in a number.
2. What is the difference between face value and place value?
Ans. Face value refers to the value of a digit itself, irrespective of its position in a number. For example, in the number 456, the face value of 5 is 5. Place value, on the other hand, is the value of a digit based on its position. In the same number, the place value of 5 is 50, as it is in the tens place. Thus, face value is the digit, while place value is the digit multiplied by its positional value.
3. What are the different types of numbers?
Ans. Numbers are categorised into several types based on their properties. The main types include natural numbers (1, 2, 3, ...), whole numbers (0, 1, 2, 3, ...), integers (..., -2, -1, 0, 1, 2, ...), rational numbers (numbers that can be expressed as a fraction, e.g., ½), and irrational numbers (numbers that cannot be expressed as a simple fraction, e.g., √2). Additionally, prime numbers (numbers greater than 1 that have no divisors other than 1 and themselves) and composite numbers (numbers that can be divided by numbers other than 1 and themselves) are also significant classifications.
4. What are some important algebraic formulae related to numbers?
Ans. Important algebraic formulae include the square of a binomial, which is given by (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b². Another significant formula is the difference of squares, which states that a² - b² = (a + b)(a - b). These formulae are fundamental in simplifying expressions and solving equations in algebra.
5. How can one determine the divisibility of a number?
Ans. The tests of divisibility are rules that help determine whether a number can be divided by another without leaving a remainder. For example, a number is divisible by 2 if its last digit is even, by 3 if the sum of its digits is divisible by 3, by 5 if it ends in either 0 or 5, and by 10 if it ends in 0. These rules facilitate quick calculations and help in identifying factors of numbers efficiently.
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