Jkssb syllabus

QUANTITATIVE APTITUDE • COMPETITIVE EXAMS

01

Number System

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Number System Roadmap

1. Basic Number Types 2. Integers & Number Line 3. Important Properties 4. Prime & Composite Numbers 5. Factors & Multiples 6. Divisibility Rules 7. HCF & LCM 8. Remainders & Modular Arithmetic 9. Unit Digit & Cyclicity 10. Factorials & Trailing Zeroes 11. Perfect Squares & Cubes 12. Fractions & Decimals 13. Digits & Place Value 14. Number Bases 15. High-Speed Shortcuts 16. Competitive Solved Questions 17. Practice Set 18. Final Revision Sheet

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Chapter Goal

Number System is the foundation of Quantitative Aptitude. The aim is not only to calculate, but to recognise the pattern quickly. Learn the definitions first, then the divisibility rules, factor methods, HCF–LCM, remainders and digit tricks. These ideas appear repeatedly in competitive examinations.

1. Basic Number Types

1.1 Natural Numbers

Natural numbers are the counting numbers used to count objects.

\[\boxed{\mathbb N=\{1,2,3,4,5,\ldots\}}\]

Exam Point: In most aptitude questions, natural numbers start from \(1\). Always check the convention if a question defines it differently.

1.2 Whole Numbers

Whole numbers are natural numbers together with zero.

\[\boxed{\mathbb W=\{0,1,2,3,4,\ldots\}}\]

1.3 Integers

Integers include negative numbers, zero and positive numbers.

\[\boxed{\mathbb Z=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}}\]

1.4 Rational Numbers

A number is rational if it can be written as \(\frac pq\), where \(p,q\) are integers and \(q\neq0\).

\[\boxed{x=\frac pq,\quad p,q\in\mathbb Z,\ q\neq0}\]

Examples: \( \frac23,\;-5,\;0,\;1.25\).

1.5 Irrational Numbers

An irrational number cannot be expressed in the form \(\frac pq\), where \(p\) and \(q\) are integers and \(q\neq0\).

Examples include \(\sqrt2,\sqrt3,\pi\).

1.6 Real Numbers

Real numbers consist of both rational and irrational numbers.

\[\boxed{\text{Real Numbers}=\text{Rational Numbers}\cup\text{Irrational Numbers}}\]

Natural1, 2, 3, ...

Whole0, 1, 2, ...

Integer..., −2, −1, 0, 1, 2, ...

Solved Example 1

Question: Which of the following is irrational: \(\frac73,\sqrt{16},\sqrt5,0\)?

1. \(\frac73\) is rational.
2. \(\sqrt{16}=4\), so it is rational.
3. \(0\) is rational because \(0=\frac01\).
4. \(\sqrt5\) cannot be expressed as a ratio of integers.
5. Answer: \(\boxed{\sqrt5}\)

2. Integers & Number Line

2.1 Positive and Negative Numbers

Numbers to the right of zero on the number line are positive, while numbers to the left are negative.

Remember: On a number line, the number farther to the right is greater.

For example, \(3>-2\) and \(-2>-5\).

2.2 Absolute Value

The absolute value of a number is its distance from zero. It is never negative.

\[\boxed{|a|\ge0}\]

Examples: \(|7|=7,\quad|-7|=7,\quad|0|=0\).

2.3 Sign Rules

Operation	Same Signs	Different Signs
Addition	Add magnitudes, keep sign	Subtract magnitudes, keep sign of larger magnitude
Multiplication	Positive	Negative
Division	Positive	Negative

Solved Example 2

Question: Simplify \(-18+7-5\).

1. \(-18+7=-11\).
2. \(-11-5=-16\).
3. Answer: \(\boxed{-16}\)

3. Important Properties of Numbers

3.1 Even and Odd Numbers

An even number is divisible by \(2\). An odd number is not divisible by \(2\).

\[\boxed{\text{Even}=2n,\qquad \text{Odd}=2n+1}\]

Operation	Result
Even + Even	Even
Odd + Odd	Even
Even + Odd	Odd
Even × any integer	Even
Odd × Odd	Odd

3.2 Consecutive Numbers

Consecutive natural numbers differ by \(1\): \(n,n+1,n+2,\ldots\).

Fast Formula: The average of consecutive numbers is the middle number when their count is odd.

3.3 Sum of First \(n\) Natural Numbers

\[\boxed{1+2+3+\cdots+n=\frac{n(n+1)}2}\]

3.4 Sum of First \(n\) Odd Numbers

\[\boxed{1+3+5+\cdots+(2n-1)=n^2}\]

3.5 Sum of First \(n\) Even Numbers

\[\boxed{2+4+6+\cdots+2n=n(n+1)}\]

3.6 Sum of Squares

\[\boxed{1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}6}\]

3.7 Sum of Cubes

\[\boxed{1^3+2^3+\cdots+n^3=\left[\frac{n(n+1)}2\right]^2}\]

Solved Example 3

Question: Find the sum of the first \(50\) natural numbers.

1. Use \(S=\frac{n(n+1)}2\).
2. Put \(n=50\).
3. \(S=\frac{50\times51}{2}=25\times51=1275\).
4. Answer: \(\boxed{1275}\)

4. Prime & Composite Numbers

4.1 Prime Number

A prime number has exactly two positive factors: \(1\) and itself.

Examples: \(2,3,5,7,11,13,17,\ldots\)

Very Important: \(1\) is neither prime nor composite.

4.2 Composite Number

A composite number has more than two positive factors.

Examples: \(4,6,8,9,10,12,\ldots\)

4.3 The Number 2

\(2\) is the only even prime number. Every other even number greater than \(2\) is composite.

4.4 Prime Factorisation

Writing a number as a product of prime numbers is called prime factorisation.

\[360=2^3\times3^2\times5\]

Solved Example 4

Question: Find the prime factorisation of \(840\).

1. \(840=84\times10\).
2. \(84=2^2\times3\times7\).
3. \(10=2\times5\).
4. Therefore \(840=2^3\times3\times5\times7\).
5. Answer: \(\boxed{2^3\times3\times5\times7}\)

5. Factors & Multiples

5.1 Factor

A number \(a\) is a factor of \(N\) if \(N\) is exactly divisible by \(a\).

For \(12\), the positive factors are \(1,2,3,4,6,12\).

5.2 Multiple

A multiple of \(a\) is obtained by multiplying \(a\) by an integer.

Multiples of \(5\): \(5,10,15,20,25,\ldots\)

5.3 Number of Factors

If

\[N=p_1^{a}p_2^{b}p_3^{c}\cdots\]

where \(p_1,p_2,p_3,\ldots\) are distinct primes, then the number of positive factors is:

\[\boxed{(a+1)(b+1)(c+1)\cdots}\]

Solved Example 5

Question: How many positive factors does \(360\) have?

1. Prime factorise: \(360=2^3\times3^2\times5^1\).
2. Apply the formula: \((3+1)(2+1)(1+1)\).
3. \(=4\times3\times2=24\).
4. Answer: \(\boxed{24}\)

5.4 Sum of Positive Factors

If \(N=p^a q^b r^c\), then the sum of all positive divisors is:

\[ \boxed{\sigma(N)= \left(1+p+p^2+\cdots+p^a\right) \left(1+q+q^2+\cdots+q^b\right) \left(1+r+\cdots+r^c\right)} \]

5.5 Odd and Even Number of Factors

Key Result: A positive integer has an odd number of factors if and only if it is a perfect square.

Reason: factors normally occur in pairs \((d,N/d)\). Only for a square does one factor pair meet at \(\sqrt N\).

6. Divisibility Rules — Must Learn

Divisor	Quick Test
2	Last digit is even: 0, 2, 4, 6 or 8.
3	Sum of digits is divisible by 3.
4	Last two digits are divisible by 4.
5	Last digit is 0 or 5.
6	Number is divisible by both 2 and 3.
7	Double the last digit and subtract from the remaining number; repeat if needed.
8	Last three digits are divisible by 8.
9	Sum of digits is divisible by 9.
10	Last digit is 0.
11	Difference between alternating digit sums is 0 or a multiple of 11.
12	Divisible by both 3 and 4.
15	Divisible by both 3 and 5.
18	Divisible by both 2 and 9.
20	Last two digits are divisible by 20.
25	Last two digits are 00, 25, 50 or 75.
125	Last three digits are divisible by 125.

Solved Example 6

Question: Is \(7,425\) divisible by \(3\), \(5\), \(9\) and \(11\)?

1. Digit sum \(=7+4+2+5=18\), so it is divisible by \(3\) and \(9\).
2. Last digit is \(5\), so it is divisible by \(5\).
3. For \(11\): \((7+2)-(4+5)=9-9=0\), so it is divisible by \(11\).
4. Answer: It is divisible by 3, 5, 9 and 11.

7. HCF & LCM

7.1 HCF / GCD

HCF means Highest Common Factor. It is the greatest positive number that divides all the given numbers exactly.

7.2 LCM

LCM means Least Common Multiple. It is the smallest positive number that is a multiple of all the given numbers.

7.3 Prime Factor Method

For HCF, take the smallest powers of common prime factors.

For LCM, take the greatest powers of every prime factor appearing.

\[\boxed{\text{For two positive integers }a,b:\quad \mathrm{HCF}(a,b)\times\mathrm{LCM}(a,b)=a\times b}\]

Solved Example 7

Question: Find HCF and LCM of \(72\) and \(120\).

1. \(72=2^3\times3^2\).
2. \(120=2^3\times3\times5\).
3. HCF \(=2^3\times3=24\).
4. LCM \(=2^3\times3^2\times5=360\).
5. Check: \(24\times360=8640=72\times120\).
6. Answer: HCF \(=\boxed{24}\), LCM \(=\boxed{360}\).

7.4 Product and HCF-LCM Problems

Shortcut: If HCF and LCM of two numbers are \(h\) and \(l\), and one number is \(a\), then the other number is \[ \boxed{\frac{hl}{a}} \] provided the given values are consistent.

8. Remainders & Modular Arithmetic

8.1 Division Algorithm

When a positive integer \(N\) is divided by a positive divisor \(d\), we can write:

\[\boxed{N=dq+r,\quad 0\le r

Here \(q\) is the quotient and \(r\) is the remainder.

8.2 Remainder of a Sum

If two numbers leave remainders \(r_1,r_2\) on division by \(m\), then their sum leaves the remainder of \(r_1+r_2\) after reducing it modulo \(m\).

\[ \boxed{(a+b)\bmod m=[(a\bmod m)+(b\bmod m)]\bmod m} \]

8.3 Remainder of a Product

\[ \boxed{(ab)\bmod m=[(a\bmod m)(b\bmod m)]\bmod m} \]

Solved Example 8

Question: Find the remainder when \(37\times48\) is divided by \(5\).

1. \(37\) leaves remainder \(2\) on division by \(5\).
2. \(48\) leaves remainder \(3\).
3. So required remainder \(=2\times3=6\).
4. \(6\) leaves remainder \(1\) when divided by \(5\).
5. Answer: \(\boxed{1}\)

8.4 Negative Remainders

In standard remainder questions, the remainder is taken between \(0\) and \(m-1\).

For example, \(-3\equiv 4\pmod7\), because \(-3+7=4\).

8.5 Remainder of Large Powers

For \(a^n\), first find the repeating pattern of remainders modulo the required divisor.

Solved Example 9

Question: Find the remainder when \(2^{10}\) is divided by \(7\).

1. Powers of \(2\) modulo \(7\): \(2,4,1,2,4,1,\ldots\).
2. The cycle length is \(3\).
3. \(10=3\times3+1\), so use the first value of the cycle.
4. Therefore \(2^{10}\equiv2\pmod7\).
5. Answer: \(\boxed{2}\)

9. Unit Digit & Cyclicity

9.1 Why Cycles Matter

For large powers, the unit digit usually repeats in a cycle. We only need the position of the exponent inside that cycle.

Base	Unit-Digit Cycle	Cycle Length
2	2, 4, 8, 6	4
3	3, 9, 7, 1	4
4	4, 6	2
5	5	1
6	6	1
7	7, 9, 3, 1	4
8	8, 4, 2, 6	4
9	9, 1	2

Fast Rule: For a cycle of length \(k\), calculate \(n\bmod k\). If the remainder is \(0\), use the \(k\)-th item of the cycle.

Solved Example 10

Question: Find the unit digit of \(7^{103}\).

1. Cycle for \(7\): \(7,9,3,1\).
2. Cycle length \(=4\).
3. \(103\div4\) leaves remainder \(3\).
4. Use the 3rd item of the cycle: \(3\).
5. Answer: \(\boxed{3}\)

9.2 Unit Digit of a Product

Find the unit digit of each factor or power first, multiply those unit digits, and retain only the final digit.

10. Factorials & Trailing Zeroes

10.1 Factorial

\[\boxed{n!=n(n-1)(n-2)\cdots3\cdot2\cdot1}\]

Also, \(0!=1\).

10.2 Trailing Zeroes in \(n!\)

A trailing zero comes from a factor \(10=2\times5\). In factorials, factors of \(2\) are more common than factors of \(5\). Therefore, count the factors of \(5\).

\[ \boxed{\text{Number of trailing zeroes in }n! =\left\lfloor\frac n5\right\rfloor+ \left\lfloor\frac n{25}\right\rfloor+ \left\lfloor\frac n{125}\right\rfloor+\cdots} \]

Solved Example 11

Question: How many trailing zeroes are there in \(100!\)?

1. \(\left\lfloor100/5\right\rfloor=20\).
2. \(\left\lfloor100/25\right\rfloor=4\).
3. \(\left\lfloor100/125\right\rfloor=0\).
4. Total \(=20+4=24\).
5. Answer: \(\boxed{24}\)

Common Mistake: Do not simply calculate \(n/5\) for large factorials. Higher powers such as \(25,125,625,\ldots\) contribute additional factors of \(5\).

11. Perfect Squares & Cubes

11.1 Perfect Square

A perfect square is the square of an integer.

\[\boxed{N=k^2}\]

11.2 Last Digit of a Perfect Square

A perfect square can end only in \(0,1,4,5,6,\) or \(9\).

Therefore: A number ending in \(2,3,7,\) or \(8\) cannot be a perfect square.

11.3 Prime Factor Condition for a Perfect Square

In the prime factorisation of a perfect square, every exponent must be even.

\[\boxed{N\text{ is a perfect square }\iff\text{all prime exponents are even}}\]

11.4 Perfect Cube

In the prime factorisation of a perfect cube, every exponent must be a multiple of \(3\).

\[\boxed{N\text{ is a perfect cube }\iff\text{all prime exponents are multiples of }3}\]

Solved Example 12

Question: Is \(3600\) a perfect square?

1. \(3600=36\times100=2^4\times3^2\times5^2\).
2. All exponents \(4,2,2\) are even.
3. Therefore, \(3600\) is a perfect square.
4. Indeed, \(3600=60^2\).
5. Answer: Yes, \(\boxed{60^2}\)

12. Fractions & Decimals

12.1 Proper and Improper Fractions

A proper fraction has numerator smaller than denominator. An improper fraction has numerator greater than or equal to denominator.

12.2 Terminating Decimal

After reducing a rational fraction to lowest terms, its decimal terminates if the denominator contains no primes other than \(2\) and \(5\).

\[ \boxed{\text{Denominator}=2^a5^b} \]

12.3 Recurring Decimal

If the reduced denominator contains any prime factor other than \(2\) or \(5\), the decimal expansion is recurring.

Solved Example 13

Question: Decide whether \(\frac{7}{40}\) has a terminating decimal expansion.

1. \(40=2^3\times5\).
2. The denominator contains only \(2\) and \(5\).
3. Therefore the decimal terminates.
4. \(\frac7{40}=0.175\).
5. Answer: Terminating

12.4 Comparing Fractions

For positive fractions \(\frac ab\) and \(\frac cd\), cross multiplication can be used:

\[ \boxed{\frac ab>\frac cd\iff ad>bc}\qquad(b,d>0) \]

13. Digits & Place Value

13.1 Two-Digit Number

If tens digit is \(a\) and units digit is \(b\), the number is:

\[\boxed{10a+b}\]

13.2 Three-Digit Number

If digits are \(a,b,c\), the number is:

\[\boxed{100a+10b+c}\]

13.3 Number Reversal

For a two-digit number \(10a+b\), its reverse is \(10b+a\).

\[ \boxed{(10a+b)-(10b+a)=9(a-b)} \]

Competitive Trick: The difference between a two-digit number and its reverse is always divisible by \(9\).

13.4 Digit Sum and Modulo 9

A number and the sum of its digits have the same remainder when divided by \(9\).

\[ \boxed{N\equiv\text{sum of digits of }N\pmod9} \]

Solved Example 14

Question: A two-digit number has digits \(a\) and \(b\). Its reverse is 27 greater than the original number. Find the difference between the digits.

1. Original \(=10a+b\), reverse \(=10b+a\).
2. Given \(10b+a-(10a+b)=27\).
3. \(9(b-a)=27\).
4. \(b-a=3\).
5. Answer: \(\boxed{3}\)

14. Number Bases

14.1 What is a Base?

A number system with base \(b\) uses digits from \(0\) to \(b-1\).

For example, binary has base \(2\), so it uses only \(0\) and \(1\).

14.2 Decimal Expansion of a Number in Base \(b\)

If a number is written as \(abc_b\), its decimal value is:

\[ \boxed{abc_b=a b^2+b b+c} \]

14.3 Binary to Decimal

Solved Example 15

Question: Convert \((1011)_2\) into decimal.

1. Use place values \(2^3,2^2,2^1,2^0\).
2. \((1011)_2=1(8)+0(4)+1(2)+1(1)\).
3. \(=8+2+1=11\).
4. Answer: \((1011)_2=(11)_{10}\)

15. High-Speed Competitive Shortcuts

15.1 Divisibility Combination

If two divisors are coprime, divisibility by both is equivalent to divisibility by their product.

Example: divisible by \(3\) and \(5\) \(\Rightarrow\) divisible by \(15\).

15.2 Consecutive Integers

The product of two consecutive integers is always even:

\[\boxed{n(n+1)\text{ is divisible by }2}\]

The product of three consecutive integers is always divisible by \(6\):

\[\boxed{n(n+1)(n+2)\text{ is divisible by }6}\]

15.3 Difference of Squares

\[\boxed{a^2-b^2=(a-b)(a+b)}\]

This identity is extremely useful for mental calculation and factorisation.

15.4 Multiplication Near 100

For \(98\times97\):

\(98=100-2,\quad97=100-3\).

\[ 98\times97=(100-2)(100-3)=10000-500+6=9506. \]

15.5 Divisibility by 9 Using Digit Sum

Instead of dividing a large number by \(9\), repeatedly add its digits.

\[ 987654\rightarrow9+8+7+6+5+4=39\rightarrow3+9=12\rightarrow1+2=3 \]

Therefore the number is not divisible by \(9\).

15.6 Count Numbers Divisible by \(k\)

From \(1\) to \(N\), the number of multiples of \(k\) is:

\[ \boxed{\left\lfloor\frac Nk\right\rfloor} \]

From \(A\) to \(B\), inclusive:

\[ \boxed{\left\lfloor\frac Bk\right\rfloor-\left\lfloor\frac{A-1}{k}\right\rfloor} \]

16. Competitive Exam Solved Questions

Question 1 — Number of Factors

How many factors does \(720\) have?

\[ 720=2^4\times3^2\times5^1 \]

Number of factors: \[ (4+1)(2+1)(1+1)=5\times3\times2=30. \]

Answer: \(\boxed{30}\)

Question 2 — HCF

Find the HCF of \(144,216\) and \(288\).

\(144=2^4\times3^2\)

\(216=2^3\times3^3\)

\(288=2^5\times3^2\)

Take minimum powers: \(2^3\times3^2=8\times9=72\).

Answer: \(\boxed{72}\)

Question 3 — Unit Digit

Find the unit digit of \(3^{202}\).

Cycle of \(3\): \(3,9,7,1\), length \(4\).

\(202\bmod4=2\).

The second number in the cycle is \(9\).

Answer: \(\boxed9\)

Question 4 — Remainder

Find the remainder when \(2^{15}\) is divided by \(7\).

\(2^3=8\equiv1\pmod7\).

Since \(15=3\times5\),

\[ 2^{15}=(2^3)^5\equiv1^5\equiv1\pmod7. \]

Answer: \(\boxed1\)

Question 5 — Trailing Zeroes

Find the number of trailing zeroes in \(125!\).

\[ \left\lfloor\frac{125}{5}\right\rfloor+ \left\lfloor\frac{125}{25}\right\rfloor+ \left\lfloor\frac{125}{125}\right\rfloor =25+5+1=31. \]

Answer: \(\boxed{31}\)

Question 6 — Perfect Square

Which smallest number should multiply \(180\) to make it a perfect square?

\[ 180=2^2\times3^2\times5 \]

Only exponent of \(5\) is odd.

Multiply by \(5\): \(180\times5=900=30^2\).

Answer: \(\boxed5\)

Question 7 — Divisibility

Find the digit \(x\) so that \(53x4\) is divisible by \(9\).

Digit sum \(=5+3+x+4=12+x\).

For divisibility by \(9\), \(12+x\) must be a multiple of \(9\).

The next multiple is \(18\), so \(x=6\).

Answer: \(\boxed6\)

Question 8 — Consecutive Numbers

Find the sum of five consecutive integers whose middle number is \(27\).

The numbers are \(25,26,27,28,29\).

Sum \(=25+26+27+28+29=135\).

Or directly: average \(=27\), count \(=5\), so sum \(=27\times5=135\).

Answer: \(\boxed{135}\)

Question 9 — Count Multiples

How many numbers from \(1\) to \(500\) are divisible by \(12\)?

\[ \left\lfloor\frac{500}{12}\right\rfloor=41. \]

Answer: \(\boxed{41}\)

Question 10 — Digit Reversal

A two-digit number exceeds its reverse by \(45\). Find the difference between its digits.

Difference \(=9(a-b)\).

So \(9(a-b)=45\).

Therefore \(a-b=5\).

Answer: \(\boxed5\)

17. Practice Set — Competitive Level

Try Without Looking at the Answers

1. Find the HCF of \(84\) and \(126\).
2. Find the LCM of \(18,24\).
3. How many factors does \(540\) have?
4. Find the unit digit of \(8^{57}\).
5. Find the remainder when \(7^{20}\) is divided by \(6\).
6. How many trailing zeroes are there in \(75!\)?
7. Find the smallest number by which \(72\) should be multiplied to make it a perfect square.
8. Find the digit \(x\) if \(72x6\) is divisible by \(9\).
9. How many multiples of \(7\) lie between \(100\) and \(500\), inclusive?
10. Find the sum of the first \(40\) natural numbers.
11. Determine whether \(\frac{13}{80}\) has a terminating decimal expansion.
12. Find the unit digit of \(9^{999}\).
13. Find the number of positive divisors of \(2^5\times3^2\times7\).
14. Find the HCF and LCM of \(48\) and \(180\).
15. Find the remainder when \(123456\) is divided by \(9\).

Answer Key

Q	Answer	Q	Answer	Q	Answer
1	42	6	18	11	Terminating
2	72	7	2	12	9
3	24	8	3	13	36
4	2	9	57	14	HCF 12, LCM 720
5	1	10	820	15	3

18. Final Revision Sheet

Number Types

Natural: \(1,2,3,\ldots\)

Whole: \(0,1,2,3,\ldots\)

Integers: negative, zero and positive whole numbers.

Rational: \(\frac pq,\;q\neq0\).

Irrational: cannot be written as \(\frac pq\).

Real = rational + irrational.

Must-Remember Formulas

\[ \boxed{1+2+\cdots+n=\frac{n(n+1)}2} \]

\[ \boxed{1+3+\cdots+(2n-1)=n^2} \]

\[ \boxed{1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}6} \]

\[ \boxed{\text{Number of factors of }p^aq^br^c=(a+1)(b+1)(c+1)} \]

\[ \boxed{\mathrm{HCF}\times\mathrm{LCM}=a\times b} \]

\[ \boxed{N=dq+r,\quad0\le r

\[ \boxed{\text{Trailing zeroes in }n!=\sum_{k\ge1}\left\lfloor\frac{n}{5^k}\right\rfloor} \]

Last-Minute Exam Checklist

• Check whether the question asks for factor, multiple, HCF or LCM.
• For divisibility, use the shortest applicable rule instead of long division.
• For large powers, look for a cycle.
• For factorial zeroes, count powers of \(5\), not just \(n/5\).
• For factor-count questions, first write the prime factorisation.
• For perfect squares, check whether every prime exponent is even.
• For a two-digit number, write it as \(10a+b\).
• Always reduce a fraction before deciding whether its decimal terminates.
• In remainder questions, keep every remainder between \(0\) and divisor minus \(1\).
• Do not spend one minute on a calculation that can be solved by a pattern in ten seconds.

Smart Strategy

First master divisibility + prime factorisation + HCF/LCM. Then master remainders + unit digits + factorials. These areas give you the biggest speed advantage in Number System questions.

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