🔢 Number System, HCF & LCM, Divisibility

Day 1 | ⏱️ Part of 4-Hour Quant Block | TCS Interview Prep

📌 Subtopics

• Number System: Types of numbers (Natural, Whole, Integers, Rational, Irrational, Real, Complex)
• HCF (Highest Common Factor): Euclidean algorithm, prime factorization method
• LCM (Least Common Multiple): Division method, prime factorization
• Divisibility Rules: Rules for 2, 3, 4, 5, 6, 7, 8, 9, 11, and composite numbers
• Rational & Irrational Numbers
• Simple Arithmetic Operations

📖 Comprehensive Notes

1. Types of Numbers (Number System)

Numbers are classified based on their mathematical properties:

• Natural Numbers ($\mathbb{N}$): Counting numbers starting from $1$ ($1, 2, 3, \dots$).
• Whole Numbers ($\mathbb{W}$): Natural numbers including zero ($0, 1, 2, 3, \dots$).
• Integers ($\mathbb{Z}$): Whole numbers and their negative counterparts ($\dots, -2, -1, 0, 1, 2, \dots$).
• Rational Numbers ($\mathbb{Q}$): Numbers that can be expressed as $\frac{p}{q}$ where $p, q \in \mathbb{Z}$ and $q \neq 0$ (e.g., $\frac{3}{4}$, $5$, $-0.25$).
• Irrational Numbers: Numbers that cannot be expressed as $\frac{p}{q}$. Their decimal expansions are non-terminating and non-recurring (e.g., $\sqrt{2}$, $\pi$, $e$).
• Real Numbers ($\mathbb{R}$): The combination of rational and irrational numbers.
• Prime Numbers: Natural numbers $> 1$ with exactly two factors: $1$ and itself (e.g., $2, 3, 5, 7, 11$). $2$ is the only even prime.
• Composite Numbers: Natural numbers $> 1$ that have more than two factors (e.g., $4, 6, 8, 9$).

2. Highest Common Factor (HCF)

https://www.geeksforgeeks.org/aptitude/hcf-and-lcm-aptitude-questions/ HCF (also called GCD - Greatest Common Divisor) of two or more numbers is the largest number that divides each of them without leaving a remainder.

• Prime Factorization Method: Express numbers as product of prime powers; multiply the lowest power of all common prime factors.
• Euclidean Algorithm (Division Method): Keep dividing the larger number by the smaller number until the remainder is $0$. The last divisor is the HCF.
• HCF of Fractions: $\text{HCF of Fractions} = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$

Example: Find HCF of $72$ and $120$. $72 = 2^3 \times 3^2$ $120 = 2^3 \times 3^1 \times 5^1$ Common prime factors with lowest powers: $2^3 \times 3^1 = 8 \times 3 = 24$.

3. Least Common Multiple (LCM)

LCM of two or more numbers is the smallest number that is a multiple of each of them.

• Prime Factorization Method: Multiply the highest power of all prime factors involved.
• LCM of Fractions: $\text{LCM of Fractions} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$
• Core Property: For two numbers $a$ and $b$: $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$

Example: Find LCM of $72$ and $120$. $72 = 2^3 \times 3^2$ $120 = 2^3 \times 3^1 \times 5^1$ Highest power of all prime factors: $2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 360$.

4. Divisibility Rules

Used to quickly determine if a number is divisible by another without full division:

• Divisible by 2: Last digit is even ($0, 2, 4, 6, 8$).
• Divisible by 3: Sum of digits is divisible by $3$.
• Divisible by 4: Number formed by the last two digits is divisible by $4$.
• Divisible by 5: Last digit is $0$ or $5$.
• Divisible by 6: Divisible by both $2$ and $3$.
• Divisible by 7: Double the last digit and subtract it from the remaining number; if the result is divisible by $7$, the original number is too.
• Divisible by 8: Number formed by the last three digits is divisible by $8$.
• Divisible by 9: Sum of digits is divisible by $9$.
• Divisible by 10: Last digit is $0$.
• Divisible by 11: The difference between the sum of digits at odd places and even places is either $0$ or a multiple of $11$.

Example: Is $1824$ divisible by $8$ and $11$?

• For $8$: Last 3 digits $824 / 8 = 103$ (remainder 0). Yes.
• For $11$: Sum of odd places ($1+2=3$), sum of even places ($8+4=12$). Difference: $|3 - 12| = 9$ (not $0$ or a multiple of $11$). No.

🌐 Learning Resources

✅ Practice Links

📝 Key Formulas

HCF × LCM = Product of the two numbers
LCM(a, b) = (a × b) / HCF(a, b)

• Divisibility by 3: Sum of digits divisible by 3
• Divisibility by 4: Last 2 digits divisible by 4
• Divisibility by 8: Last 3 digits divisible by 8
• Divisibility by 11: Alternating sum of digits divisible by 11

🔗 Related Topics

• Equations — Solving equations often requires HCF/LCM concepts
• Ratios and Proportion — Ratios use divisibility and LCM
• Greedy Number Theory and Bit Manipulation — GCD/LCM algorithms in coding context
• Percentages — Percentage calculations build on number fundamentals
• Ages — Age problems reduce to linear equations with number properties