Mathematics

Number System & Basics — Foundation of Every Math Paper

The Number System is the foundation of every objective and descriptive math paper. SSC, RRB Group D/NTPC, Banking and most state exams begin with 3–5 questions that directly test fractions, decimals, divisibility, LCM/HCF, BODMAS and roots. Master this section first — it makes every later topic faster.

Exam relevance: Expect 4–6 marks in SSC CGL/CHSL, 3–4 marks in RRB Group D, and 2–3 marks in IBPS Clerk. Divisibility shortcuts and BODMAS speed are the fastest scoring areas.

On this page

  1. Number System (Fractions, Decimals, Integers, Divisibility)
  2. LCM and HCF (Least Common Multiple & Highest Common Factor)
  3. BODMAS / PEMDAS (Order of Operations)
  4. Square Roots & Cube Roots

1Number System (Fractions, Decimals, Integers, Divisibility)

The number system classifies numbers into Natural (1, 2, 3…), Whole (0, 1, 2…), Integers (…-2, -1, 0, 1, 2…), Rational (p/q form), Irrational (√2, π) and Real numbers.

A fraction (a/b) represents a part of a whole; decimals are the same value written with a base-10 point. Divisibility rules let you check whether a number divides another without long division: by 2 — last digit even; by 3 — sum of digits divisible by 3; by 4 — last two digits divisible by 4; by 5 — last digit 0 or 5; by 6 — divisible by both 2 and 3; by 8 — last three digits divisible by 8; by 9 — digit sum divisible by 9; by 11 — difference of alternate digit sums divisible by 11.

Examples
  • Is 7,29,540 divisible by 9? Digit sum = 7+2+9+5+4+0 = 27 → yes (27 ÷ 9 = 3).
  • Is 2,431 divisible by 11? Alternate sum = (2+3) − (4+1) = 5 − 5 = 0 → yes.
  • Convert 0.625 to fraction: 625/1000 = 5/8.
  • Place value of 7 in 4,73,592 is 70,000; face value is 7.
Exam tip: A number divisible by 12 must be divisible by both 3 and 4 (not 2 and 6). Always pick co-prime factors.

2LCM and HCF (Least Common Multiple & Highest Common Factor)

HCF is the largest number that divides two or more numbers exactly; LCM is the smallest number that is exactly divisible by them.

Use prime factorisation: HCF = product of common primes with the lowest powers; LCM = product of all primes with the highest powers. Key identity: HCF × LCM = product of the two numbers (works for any two numbers). For fractions: HCF = HCF(numerators) ÷ LCM(denominators); LCM = LCM(numerators) ÷ HCF(denominators).

Examples
  • HCF and LCM of 12 and 18: 12 = 2²×3, 18 = 2×3² → HCF = 2×3 = 6, LCM = 2²×3² = 36.
  • Verify: 6 × 36 = 216 = 12 × 18. ✓
  • Largest 4-digit number divisible by 15, 20 and 24: LCM = 120 → 9999/120 → 9960.
  • HCF of 2/3 and 4/9 = HCF(2,4)/LCM(3,9) = 2/9.
Exam tip: When a problem asks for the largest tile/measure that fits exactly, use HCF. When it asks for the smallest distance/time/quantity that contains all given quantities a whole number of times, use LCM.

3BODMAS / PEMDAS (Order of Operations)

BODMAS is the rule for the order in which a mixed expression is simplified: Brackets → Of → Division → Multiplication → Addition → Subtraction.

Brackets are solved innermost first: (), {}, []. 'Of' means multiplication but is solved before regular division and multiplication. Division and multiplication share the same level — solve left to right. Addition and subtraction also share the same level — solve left to right. Always replace 'is to' or 'of' literally before pulling out signs.

Examples
  • Simplify: 12 + 6 ÷ 2 × 3 − 4 = 12 + (6 ÷ 2) × 3 − 4 = 12 + 9 − 4 = 17.
  • Simplify: 1/2 of 60 ÷ 3 + 5 = (30 ÷ 3) + 5 = 10 + 5 = 15.
  • Simplify: [40 − {6 + 2 × (8 ÷ 4)}] = [40 − {6 + 4}] = 30.
  • Simplify: 5 + 3 × (2² − 1) = 5 + 3 × 3 = 14.
Exam tip: A negative sign before a bracket flips every sign inside when the bracket is removed: −(a − b + c) = −a + b − c.

4Square Roots & Cube Roots

The square root (√n) of a number is the value which when multiplied by itself gives the original number. The cube root (∛n) is the value which when cubed gives the original number.

Memorise squares up to 30 and cubes up to 20 — most exam values are derived from these. For long-division square root, group digits in pairs from the decimal point. For surds: √a × √b = √(ab); √a / √b = √(a/b); (√a + √b)² = a + b + 2√(ab). Rationalise denominators by multiplying with the conjugate.

Examples
  • √1296 = 36 (since 36² = 1296).
  • ∛1331 = 11 (since 11³ = 1331).
  • Simplify √50 + √32 = 5√2 + 4√2 = 9√2.
  • Rationalise 1/(√3 − 1) → (√3 + 1)/2.
Exam tip: For perfect squares, the unit digit is always 0, 1, 4, 5, 6 or 9 — never 2, 3, 7 or 8. Use this to eliminate options instantly.

Short Tricks & Shortcuts

Use these speed tricks in the exam. Each trick is followed by a worked example so you can verify the shortcut yourself.

Trick 1Divisibility by 7 (double-and-subtract)

Double the last digit, subtract from the rest. Repeat. If the result is divisible by 7 (or is 0), the original number is too.

Example: Check 1,148: 114 − (2×8) = 98 → 9 − (2×8) = −7 → divisible by 7. ✓
Trick 2Divisibility by 11 (alternate-digit-sum)

Take the difference between the sum of digits in odd positions and the sum of digits in even positions. If it is 0 or a multiple of 11, the number is divisible by 11.

Example: For 9,16,3,82: (9+6+8) − (1+3+2) = 23 − 6 = 17 → not divisible by 11.
Trick 3Square of any number ending in 5

For a number ending in 5 (say n5): square = (n × (n+1)) followed by 25.

Example: 75² → 7 × 8 = 56 → answer 5,625. 95² → 9 × 10 = 90 → answer 9,025.
Trick 4Quick LCM-of-two-numbers

LCM(a, b) = (a × b) / HCF(a, b). Find HCF first by Euclid's algorithm — it is much faster than prime factorisation for big numbers.

Example: LCM(48, 60): HCF(48,60)=12 → LCM = (48×60)/12 = 240.
Trick 5Unit-digit elimination for perfect squares

Squares can only end in 0, 1, 4, 5, 6, 9. If an option ends in 2, 3, 7 or 8 — eliminate it instantly.

Example: Which is a perfect square: 1,728 / 1,764? 1728 ends in 8 → not a square. Answer: 1764 (= 42²).

Quick Revision Facts

  • Sum of first n natural numbers = n(n+1)/2.
  • Sum of squares of first n naturals = n(n+1)(2n+1)/6.
  • Sum of cubes of first n naturals = [n(n+1)/2]².
  • 0 is even; 1 is neither prime nor composite; 2 is the only even prime.

Frequently Asked Questions

Double the last digit and subtract it from the rest. If the result is divisible by 7 (or 0), the original is too. e.g. 343 → 34 − (2×3) = 28, divisible by 7 ✓.

It is true only for TWO numbers. For three or more numbers, the identity does not hold in general.
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