6 Golden Rules for Number System

Table of Contents
1. Introduction: Number System (GMAT)
2. Classification of Numbers & Sets
3. Divisibility & Remainders
4. HCF (GCD) & LCM
5. Cyclicity & Unit Digit Patterns
6. Factorials & Trailing Zeroes
7. Base Systems & Number Line
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Introduction: Number System (GMAT)

The Number System forms the backbone of GMAT Quant questions. Understanding types of numbers, their properties, and relationships is key to solving problems on divisibility, remainders, HCF/LCM, factorials, cyclicity, and base conversions. Mastering core rules and patterns allows you to solve questions quickly and accurately. The 6 Golden Rules below summarize the essentials for fast and effective problem-solving.

1. Classification of Numbers & Sets

Rule: Know the hierarchy and properties of all number sets.

Details:

• Natural Numbers (N): Counting numbers starting from 1 → {1,2,3,...}

• Whole Numbers (W): Natural numbers + 0 → {0,1,2,...}

• Integers (Z): Whole numbers + negatives → {...,-2,-1,0,1,2,...}

• Rational Numbers (Q): Numbers that can be expressed as p/q where p and q are integers, q ≠ 0. Example: 1/2, -7/3, 0.75

• Irrational Numbers: Cannot be expressed as p/q. Decimal is non-terminating and non-repeating. Example: √2, π

• Real Numbers (R): Rational ∪ Irrational

Important GMAT Notes:

• 1 is neither prime nor composite.

• Prime numbers have exactly 2 factors; composites have more than 2.

• Even/Odd: Even divisible by 2; odd not divisible by 2.

Mini Example:

• Which set does -3.5 belong to? → Rational Numbers

2. Divisibility & Remainders

Rule: Quickly test divisibility and calculate remainders without full division.

Divisibility Rules:

• 2 → last digit even

• 3 → sum of digits divisible by 3

• 4 → last 2 digits divisible by 4

• 5 → last digit 0 or 5

• 6 → divisible by 2 & 3

• 9 → sum of digits divisible by 9

• 10 → last digit 0

Remainder Formula:

Dividend=(Divisor×Quotient)+Remainder,0≤R<Divisor

Mini Example:

Find remainder when 234 is divided by 9: sum of digits = 2+3+4=9 → divisible → remainder = 0

3. HCF (GCD) & LCM

Rule: Use prime factorisation and the formula for fast calculation.

• HCF (GCD): Multiply all common prime factors with smallest powers

• LCM: Multiply all prime factors with highest powers

• Relationship:

$a\times b=HCF(a,b)\times LCM(a,b)a\; \times \; b\; =\; HCF(a,b)\; \times \; LCM(a,b)$a×b=HCF(a,b)×LCM(a,b)

• Co-prime: HCF(a,b) = 1; if numbers are co-prime and divisible by some x, then divisible by their product × x.

Mini Example:

• HCF of 12 & 18 → 12=2²×3¹, 18=2¹×3² → HCF = 2¹×3¹ = 6

• LCM = 2²×3² = 36

4. Cyclicity & Unit Digit Patterns

Rule: The last digit of powers of numbers follows a repeatable pattern.

Unit-digit cycles:

• 0,1,5,6 → cycle 1 (last digit same)

• 4,9 → cycle 2

• 2,3,7,8 → cycle 4

Use this for finding last digits of large powers or remainders modulo 10.

Mini Example:

• Last digit of 7⁴: Cycle of 7 → 7,9,3,1 → 7⁴ last digit = 1

• Last digit of 2¹⁰⁰: 2,4,8,6 cycle → 100 mod 4 = 0 → last digit = 6

5. Factorials & Trailing Zeroes

Rule: Trailing zeroes come from factors of 10 (2×5).

• Formula to count trailing zeroes in n!

$Number of zeroes=\lfloor \frac{n}{5}\rfloor +\lfloor \frac{n}{25}\rfloor +\lfloor \frac{n}{125}\rfloor +...\backslash text\{Number\; of\; zeroes\}\; =\; \backslash left\backslash lfloor\; \backslash frac\{n\}\{5\}\; \backslash right\backslash rfloor\; +\; \backslash left\backslash lfloor\; \backslash frac\{n\}\{25\}\; \backslash right\backslash rfloor\; +\; \backslash left\backslash lfloor\; \backslash frac\{n\}\{125\}\; \backslash right\backslash rfloor\; +\; \backslash dots$Number of zeroes=⌊5n​⌋+⌊25n​⌋+⌊125n​⌋+...

• Only count multiples of 5 (2s are always more).

Mini Example:

• Trailing zeroes in 100! = ⌊100/5⌋ + ⌊100/25⌋ = 20 + 4 = 24

6. Base Systems & Number Line

Rule: Be comfortable with base conversions and number line properties.

• Base Conversion:

  $Number=\sum (Digit\times BasePosition)\backslash text\{Number\}\; =\; \backslash sum\; (\backslash text\{Digit\}\; \times \; \backslash text\{Base\}^\{\backslash text\{Position\}\})$Number=∑(Digit×BasePosition)

• Common bases: Binary (2), Octal (8), Decimal (10), Hexadecimal (16)

• Number Line: Distance between a & b → |a-b|

• Absolute value: |a| = a if a ≥ 0, = -a if a < 0

• Perfect Numbers: Sum of proper divisors = number (6, 28...)

Mini Example:

• Convert 1011₂ to decimal → 1×2³ + 0×2² + 1×2¹ +1×2⁰ = 8+0+2+1=11