Topics covered (19)

1. Number System — Introduction to Number System The Number System is the foundation of all quantitative aptitude. Numbers are classified into Natural Numbers (1, 2, 3, …), Whole Numbers (0, 1, 2,…
2. Percentage — Understanding Percentage Percentage literally means "per hundred." It is a way of expressing a number as a fraction of 100. The symbol % is used to denote percentage. Percentages…
3. Profit & Loss — Basic Concepts of Profit and Loss Cost Price (CP) is the price at which an article is purchased. Selling Price (SP) is the price at which it is sold. If SP CP, there is a Profit;…
4. Simple & Compound Interest — Simple Interest Fundamentals Simple Interest (SI) is calculated on the original principal for the entire duration. SI = (P × R × T) / 100, where P = Principal, R = Rate of…
5. Time & Work — The Basic Concept of Time and Work The fundamental principle: if a person completes a work in n days, his 1-day work = 1/n. If A can do a work in 'a' days and B can do the same…
6. Speed, Time & Distance — Fundamental Relationship The three quantities Speed (S), Time (T), and Distance (D) are related by: D = S × T. All three are dependent on this single equation. Speed is measured…
7. Algebra — Algebraic Identities Algebraic identities are equations true for all values of variables. The most important ones for exams: (a+b)² = a²+2ab+b², (a-b)² = a²-2ab+b², a²-b² =…
8. Geometry — Lines, Angles and Triangles A straight line has 180° and a point has 360° around it. Parallel lines cut by a transversal form corresponding angles (equal), alternate interior…
9. Trigonometry — Trigonometric Ratios and Standard Values For a right-angled triangle with angle θ: sin θ = Opposite/Hypotenuse, cos θ = Adjacent/Hypotenuse, tan θ = Opposite/Adjacent.…
10. Data Interpretation — Types of Data Interpretation Questions Data Interpretation (DI) is one of the highest-weightage topics in Banking exams (IBPS PO, SBI PO) and appears in SSC CGL Tier-II as well.…
11. Ratio & Proportion — Understanding Ratio A ratio is a comparison of two quantities of the same kind expressed as a:b or a/b. Ratios are always in their simplest form (divide both terms by their HCF).…
12. Averages — Arithmetic Mean (Simple Average) The arithmetic mean (average) of n observations x₁, x₂, …, xₙ is defined as their sum divided by the count: Average = (x₁ + x₂ + … + xₙ) / n. If…
13. Mixtures & Alligation — The Alligation Rule Alligation is a rule that enables us to find the ratio in which two or more ingredients at given prices (or concentrations) must be mixed to produce a mixture…
14. Permutation & Combination — Fundamental Counting Principle If one event can occur in m ways and a second independent event can occur in n ways, then both events together can occur in m × n ways…
15. Probability — Basic Concepts of Probability Probability measures the likelihood of an event occurring. The sample space (S) is the set of all possible outcomes of an experiment. An event (E) is…
16. Mensuration (2D & 3D) — 2D Shapes: Area and Perimeter For a triangle with sides a, b, c: Perimeter = a+b+c; Area by Heron's formula = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2. For a right triangle with…
17. Pipes & Cisterns — Basic Concepts: Filling and Emptying In pipes and cisterns problems, the tank is treated as a unit of work (filling the full tank = 1 unit of work). An inlet pipe that fills the…
18. Boats & Streams — Basic Terminology and Concept In boats and streams problems, the boat's speed in still water is u (also written as B), and the stream (current) speed is v (also written as S).…
19. Statistics (Mean, Median, Mode) — Arithmetic Mean The mean of ungrouped data with n observations x₁, x₂, …, xₙ is Mean (x̄) = Σxᵢ / n. For grouped (frequency distribution) data, Mean = Σfᵢxᵢ / Σfᵢ, where fᵢ is the…

Number System

Introduction to Number System

The Number System is the foundation of all quantitative aptitude. Numbers are classified into Natural Numbers (1, 2, 3, …), Whole Numbers (0, 1, 2, 3, …), Integers (…-2, -1, 0, 1, 2,…), Rational Numbers (p/q form where q≠0), Irrational Numbers (√2, π), and Real Numbers (all of the above). Understanding these classifications is essential because exam questions often test whether a given number belongs to a particular set.

Divisibility Rules

Divisibility rules are shortcuts to determine if a number is divisible by another without performing actual division. 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 4 if the last two digits form a number divisible by 4; by 5 if it ends in 0 or 5; by 6 if divisible by both 2 and 3; by 7 if the difference between twice the last digit subtracted from the rest is divisible by 7; by 8 if the last three digits form a number divisible by 8; by 9 if the sum of digits is divisible by 9; by 11 if the alternating sum of digits is divisible by 11.

LCM and HCF

HCF (Highest Common Factor) is the greatest number that divides all given numbers exactly. LCM (Least Common Multiple) is the smallest number divisible by all given numbers. The fundamental relationship is: HCF × LCM = Product of two numbers (applicable for exactly two numbers). To find HCF using prime factorisation, take the product of all common prime factors with the lowest powers. For LCM, take the product of all prime factors with the highest powers.

Remainders and Cyclicity

The concept of remainders is heavily tested. Euler's theorem and Fermat's little theorem form the basis for remainder problems involving large powers. Cyclicity refers to the repeating pattern of unit digits of numbers raised to successive powers. For example, the unit digit of 2^n follows a cycle of 4: 2,4,8,6,2,4,8,6,… Unit digit cyclicity: 2→4, 3→4, 7→4, 8→4, 4→2, 9→2, 1,5,6→1.

Key points

• 0 is neither positive nor negative but is an even number
• 1 is neither prime nor composite
• 2 is the only even prime number
• Every natural number > 1 is either prime or composite
• The HCF of two consecutive integers is always 1
• The LCM of two co-prime numbers equals their product
• Cyclicity of unit digits: 1(1), 2(4), 3(4), 4(2), 5(1), 6(1), 7(4), 8(4), 9(2), 0(1)
• A number with an odd number of factors is always a perfect square

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