SATHEE: Number System - All Formulas & Shortcuts

Number System - All Formulas & Shortcuts

Quick reference guide for Number System with divisibility rules and remainder shortcuts

📘 Types of Numbers

1. Number Classifications

Natural Numbers (N):

1, 2, 3, 4, 5, … Counting numbers starting from 1

Whole Numbers (W):

0, 1, 2, 3, 4, 5, … Natural numbers + 0

Integers (Z):

…, -3, -2, -1, 0, 1, 2, 3, … Positive and negative whole numbers

Rational Numbers (Q):

Numbers that can be expressed as p/q where q ≠ 0 Examples: 1/2, 3/4, 0.5, 2

Irrational Numbers:

Cannot be expressed as p/q Examples: √2, π, e

2. Special Number Types

Even Numbers: 2, 4, 6, 8, 10, … (divisible by 2) Odd Numbers: 1, 3, 5, 7, 9, … (not divisible by 2)

Prime Numbers:

Numbers with exactly 2 factors (1 and itself) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, … Note: 2 is the only even prime

Composite Numbers:

Numbers with more than 2 factors 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, …

Co-prime Numbers:

Two numbers whose HCF is 1 Examples: (3,5), (8,15), (7,9)

⚡ Divisibility Rules (MUST MEMORIZE!)

3. Essential Divisibility Rules

Divisible by 2:

Last digit is 0, 2, 4, 6, or 8 Example: 1234, 5678

Divisible by 3:

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

Divisible by 4:

Last two digits divisible by 4 Example: 1216 → 16 ÷ 4 = 4 ✓

Divisible by 5:

Last digit is 0 or 5 Example: 125, 340

Divisible by 6:

Divisible by both 2 AND 3 Example: 126 (even + sum=9, div by 3) ✓

Divisible by 8:

Last three digits divisible by 8 Example: 1000 → 000 ÷ 8 = 0 ✓ Example: 2048 → 048 ÷ 8 = 6 ✓

Divisible by 9:

Sum of digits divisible by 9 Example: 729 → 7+2+9 = 18 (÷9) ✓

Divisible by 10:

Last digit is 0 Example: 1230, 4560

Divisible by 11:

Difference of sum of alternate digits = 0 or multiple of 11 Example: 1331 → (1+3) - (3+1) = 4-4 = 0 ✓ Example: 1221 → (1+2) - (2+1) = 3-3 = 0 ✓

Divisible by 12:

Divisible by both 3 AND 4

🔥 HCF & LCM

4. HCF (Highest Common Factor)

Methods to find HCF:

Prime Factorization Method:

Find prime factors of each number HCF = Product of common factors with lowest powers

Example: HCF of 12 and 18

12 = 2² × 3 18 = 2 × 3² HCF = 2¹ × 3¹ = 6

Division Method (Euclidean Algorithm):

Divide larger by smaller Continue dividing previous divisor by remainder Last divisor is HCF

5. LCM (Lowest Common Multiple)

Prime Factorization Method:

LCM = Product of all factors with highest powers

Example: LCM of 12 and 18

12 = 2² × 3 18 = 2 × 3² LCM = 2² × 3² = 4 × 9 = 36

Formula Relationship:

HCF × LCM = Product of two numbers

HCF × LCM = a × b

Example: a=12, b=18

HCF = 6, LCM = 36 6 × 36 = 216 12 × 18 = 216 ✓

💡 Remainder Theorems

6. Basic Remainder Formula

When N is divided by D:

N = D × Q + R

Where: N = Dividend D = Divisor Q = Quotient R = Remainder (0 ≤ R < D)

7. Negative Remainder Concept

Remainder when (N-a) is divisible by D:

If N leaves remainder R when divided by D, then (N-a) leaves remainder (R-a) or (R-a+D)

Example: 47 ÷ 5 leaves remainder 2

(47-3) ÷ 5 = 44 ÷ 5 leaves remainder (2-3+5) = 4

8. Wilson’s Theorem (for Primes)

If p is prime:

(p-1)! + 1 is divisible by p

Remainder of (p-1)! ÷ p = p-1 OR -1

Example: p=7

6! = 720 720 + 1 = 721 = 7 × 103 ✓ 720 ÷ 7 gives remainder 6 (which is 7-1)

9. Chinese Remainder Theorem (CRT)

When N leaves remainders r₁, r₂, r₃ with divisors d₁, d₂, d₃:

If remainders follow pattern, use:

N = LCM(d₁, d₂, d₃) × k ± common_diff

Example: N ÷ 3 leaves 2, N ÷ 4 leaves 3, N ÷ 5 leaves 4

Pattern: Each remainder = divisor - 1 N = LCM(3,4,5) × k - 1 N = 60k - 1 N = 59, 119, 179, … (k=1,2,3,…)

📊 Quick Shortcuts

Shortcut 1: Sum of First n Natural Numbers

Sum = n(n+1)/2

Example: 1+2+3+…+100

= 100 × 101/2 = 5050

Shortcut 2: Sum of First n Even Numbers

Sum = n(n+1)

Example: 2+4+6+…+100 (50 terms)

= 50 × 51 = 2550

Shortcut 3: Sum of First n Odd Numbers

Sum = n²

Example: 1+3+5+…+99 (50 terms)

= 50² = 2500

Shortcut 4: Number of Factors

If N = p₁^a × p₂^b × p₃^c:

Number of factors = (a+1)(b+1)(c+1)

Example: 72 = 2³ × 3²

Factors = (3+1)(2+1) = 4 × 3 = 12

Shortcut 5: Sum of Factors

If N = p^a (prime power):

Sum = (p^(a+1) - 1)/(p - 1)

Example: Factors of 16 = 2⁴

Sum = (2⁵ - 1)/(2-1) = 31/1 = 31 (1+2+4+8+16 = 31 ✓)

🎯 Exam Patterns

Pattern 1: Finding Last Digit

Last digit of powers:

For 2: 2, 4, 8, 6, 2, 4, 8, 6, … (cycle of 4) For 3: 3, 9, 7, 1, 3, 9, 7, 1, … (cycle of 4) For 4: 4, 6, 4, 6, … (cycle of 2) For 7: 7, 9, 3, 1, 7, 9, 3, 1, … (cycle of 4) For 8: 8, 4, 2, 6, 8, 4, 2, 6, … (cycle of 4) For 9: 9, 1, 9, 1, … (cycle of 2)

Example: Last digit of 7^123

Cycle = 4, so 123 ÷ 4 remainder = 3 3rd in cycle (7,9,3,1) = 3 Last digit = 3

Pattern 2: Largest Number Dividing

Largest number that divides (a-b), (b-c), (c-a):

= HCF of all differences

Pattern 3: Smallest Number for Remainder

Smallest number leaving remainder r when divided by a, b, c:

Number = LCM(a,b,c) × k + r Smallest = LCM + r (when k=1)

Example: Leaves 3 when divided by 5, 6, 7

LCM(5,6,7) = 210 Smallest = 210 + 3 = 213

💎 Special Formulas

10. Perfect Square Tests

Number is perfect square if:

Square root is integer
All prime factors have even powers
Ends with 0,1,4,5,6,9 (not 2,3,7,8)

Sum of first n perfect squares:

1² + 2² + 3² + … + n² = n(n+1)(2n+1)/6

11. Perfect Cube Tests

Perfect cubes end with:

0→0, 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9

Sum of first n cubes:

1³ + 2³ + 3³ + … + n³ = [n(n+1)/2]²

12. Digit Sum Formulas

Sum of digits of number N:

If N ≡ r (mod 9), then digit sum ≡ r (mod 9)

Example: 123 ÷ 9 remainder 6

Digit sum = 1+2+3 = 6 ✓

🔍 Common Mistakes

❌ Confusing HCF with LCM ❌ Forgetting 2 is the only even prime ❌ Wrong divisibility test (using 3 for 4, etc.) ❌ Not checking remainder range (0 ≤ R < D) ✅ HCF × LCM = a × b (for two numbers) ✅ Prime factorization is most reliable method ✅ Sum of digits for divisibility by 3 and 9 ✅ Last 2 digits for 4, last 3 for 8