Number System Formulas & Shortcuts

🔢 Types of Numbers

Natural Numbers (N)

N = {1, 2, 3, 4, 5, ...}
Count = n - 1 (for numbers from 1 to n)

Whole Numbers (W)

W = {0, 1, 2, 3, 4, 5, ...}
Count = n + 1 (for numbers from 0 to n)

Integers (Z)

Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Rational Numbers (Q)

Numbers that can be written as p/q where q ≠ 0

Irrational Numbers

Numbers that cannot be written as p/q form
Example: √2, √3, π

🎯 Divisibility Rules

Divisible by 2

• Rule: Last digit is even (0, 2, 4, 6, 8)

Divisible by 3

• Rule: Sum of digits is divisible by 3
• Example: 123 → 1+2+3=6 (6 ÷ 3 = 2) ✓

Divisible by 4

• Rule: Last two digits form a number divisible by 4
• Example: 1524 → 24 ÷ 4 = 6 ✓

Divisible by 5

• Rule: Last digit is 0 or 5

Divisible by 6

• Rule: Divisible by both 2 and 3

Divisible by 7

• Rule: Double the last digit, subtract from remaining
• Example: 343 → 34 - (2×3) = 28 → 28 ÷ 7 = 4 ✓

Divisible by 8

• Rule: Last three digits form a number divisible by 8
• Example: 5640 → 640 ÷ 8 = 80 ✓

Divisible by 9

• Rule: Sum of digits is divisible by 9
• Example: 891 → 8+9+1=18 (18 ÷ 9 = 2) ✓

Divisible by 10

• Rule: Last digit is 0

Divisible by 11

• Rule: Difference between sum of digits at odd and even places is 0 or multiple of 11
• Example: 2917 → (2+1) - (9+7) = 3 - 16 = -13 ✗
• Example: 91827 → (9+8+7) - (1+2) = 24 - 3 = 21 ✗
• Example: 1331 → (1+3) - (3+1) = 4 - 4 = 0 ✓

🔢 Prime and Composite Numbers

Prime Numbers

• Numbers with exactly two factors (1 and itself)
• First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
• 2 is the only even prime number

Composite Numbers

• Numbers with more than two factors
• 1 is neither prime nor composite

Test for Prime Numbers

Check divisibility from 2 to √n
If not divisible by any number → Prime

📐 LCM and HCF

Highest Common Factor (HCF/GCD)

Methods:

1. Prime Factorization Method

HCF = Product of common prime factors with lowest powers

Example: HCF of 72 and 108

• 72 = 2³ × 3²
• 108 = 2² × 3³
• HCF = 2² × 3² = 4 × 9 = 36

2. Division Method

Continue division until remainder is 0
Last non-zero remainder is HCF

Least Common Multiple (LCM)

Methods:

1. Prime Factorization Method

LCM = Product of all prime factors with highest powers

Example: LCM of 72 and 108

• 72 = 2³ × 3²
• 108 = 2² × 3³
• LCM = 2³ × 3³ = 8 × 27 = 216

2. Division Method

Continue division with prime numbers
Multiply all divisors and remaining numbers

Important Relationship

Product of two numbers = HCF × LCM

Example: For 72 and 108

• 72 × 108 = 7776
• HCF × LCM = 36 × 216 = 7776 ✓

🔢 Fractions and Decimals

Types of Fractions

• Proper Fraction: Numerator < Denominator (3/4)
• Improper Fraction: Numerator > Denominator (7/4)
• Mixed Fraction: Whole + Proper (1¾)
• Equivalent Fractions: Same value, different form (1/2 = 2/4)

Operations with Fractions

Addition/Subtraction

a/b ± c/d = (ad ± bc)/bd

Multiplication

(a/b) × (c/d) = ac/bd

Division

(a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc

Terminating and Non-terminating Decimals

Terminating Decimals

• Denominator has only 2 and 5 as prime factors
• Example: 1/8 = 0.125, 3/20 = 0.15

Non-terminating Repeating Decimals

• Denominator has prime factors other than 2 and 5
• Example: 1/3 = 0.333…, 1/7 = 0.142857…

🎯 Surds and Indices

Laws of Indices

Basic Rules

a^m × a^n = a^(m+n)
a^m ÷ a^n = a^(m-n)
(a^m)^n = a^(mn)
a^0 = 1 (a ≠ 0)
a^(-n) = 1/a^n

Fractional Indices

a^(1/n) = ⁿ√a
a^(m/n) = (ⁿ√a)^m

Simplification of Surds

Rationalizing Denominator

1/√a = √a/a
1/(√a + √b) = (√a - √b)/(a - b)
1/(√a - √b) = (√a + √b)/(a - b)

📊 Number Series Patterns

Arithmetic Progression (AP)

Formula: a, a+d, a+2d, a+3d, ...
First term: a
Common difference: d
nth term: a + (n-1)d
Sum of n terms: n/2 × [2a + (n-1)d]

Geometric Progression (GP)

Formula: a, ar, ar², ar³, ...
First term: a
Common ratio: r
nth term: a × r^(n-1)
Sum of n terms: a(r^n - 1)/(r - 1) when r ≠ 1

Special Series

Sum of first n natural numbers = n(n+1)/2
Sum of squares = n(n+1)(2n+1)/6
Sum of cubes = [n(n+1)/2]²

🧮 Shortcuts and Tricks

Square Numbers

1² = 1      11² = 121     21² = 441
2² = 4      12² = 144     22² = 484
3² = 9      13² = 169     23² = 529
4² = 16     14² = 196     24² = 576
5² = 25     15² = 225     25² = 625
6² = 36     16² = 256     26² = 676
7² = 49     17² = 289     27² = 729
8² = 64     18² = 324     28² = 784
9² = 81     19² = 361     29² = 841
10² = 100   20² = 400     30² = 900

Cube Numbers

1³ = 1      6³ = 216     11³ = 1331
2³ = 8      7³ = 343     12³ = 1728
3³ = 27     8³ = 512     13³ = 2197
4³ = 64     9³ = 729     14³ = 2744
5³ = 125    10³ = 1000   15³ = 3375

Multiplication Shortcuts

Multiply by 11

Example: 35 × 11 = 385
Step 1: 3 _ 5
Step 2: 3 (3+5) 5 = 385

Multiply by 9

Number × 9 = (Number × 10) - Number
Example: 47 × 9 = 470 - 47 = 423

Multiply by 99

Number × 99 = (Number × 100) - Number
Example: 67 × 99 = 6700 - 67 = 6633

📈 Unit Digit Patterns

Cyclicity of Unit Digits

2, 3, 7, 8: Cycle of 4
4, 9: Cycle of 2
0, 1, 5, 6: Always same

Example: Find unit digit of 7²³

• 7 has cycle of 4: 7, 9, 3, 1
• 23 ÷ 4 = 5 remainder 3
• 3rd position in cycle = 3
• Unit digit = 3

🎯 Important Formulas

Average of Numbers

Average = (Sum of all numbers)/(Number of items)

Sum of Consecutive Numbers

Sum from 1 to n = n(n+1)/2
Sum from a to b = (a + b) × (number of terms)/2

Properties of Numbers

Even + Even = Even
Odd + Odd = Even
Even + Odd = Odd

Even × Even = Even
Odd × Odd = Odd
Even × Odd = Even

📝 Practice Questions

Question 1:

Find the HCF and LCM of 84 and 144.

Solution:

• 84 = 2² × 3 × 7
• 144 = 2⁴ × 3²
• HCF = 2² × 3 = 12
• LCM = 2⁴ × 3² × 7 = 1008

Question 2:

Find the unit digit of 3⁴⁵ × 7²⁸.

Solution:

• 3 has cycle 4: 3, 9, 7, 1. 45 ÷ 4 = 11 remainder 1 → 3
• 7 has cycle 4: 7, 9, 3, 1. 28 ÷ 4 = 7 remainder 0 → 1
• 3 × 1 = 3
• Unit digit = 3

🔗 Related Topics

• Simplification - Simplification techniques
• Average - Average calculations
• Ratio and Proportion - Ratio problems
• Percentage - Percentage calculations

📚 Continue Learning