Number System - Theory & Concepts
🔢 Number System - Complete Theory
Master the foundation of all mathematics - Number System!
🎯 Types of Numbers
1. Natural Numbers (N)
N = {1, 2, 3, 4, 5, ...}
Counting numbers starting from 1
2. Whole Numbers (W)
W = {0, 1, 2, 3, 4, 5, ...}
Natural numbers + 0
3. Integers (Z)
Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
Positive and negative whole numbers
4. Even Numbers
Numbers divisible by 2
{2, 4, 6, 8, 10, ...}
Form: 2n where n ∈ N
5. Odd Numbers
Numbers NOT divisible by 2
{1, 3, 5, 7, 9, ...}
Form: 2n + 1 or 2n - 1
6. Prime Numbers
Numbers with exactly 2 factors (1 and itself)
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...}
Note: 2 is the ONLY even prime number!
Note: 1 is NOT a prime number!
7. Composite Numbers
Numbers with more than 2 factors
{4, 6, 8, 9, 10, 12, 14, 15, ...}
8. Co-prime Numbers
Two numbers whose HCF = 1
Example: (3, 8), (7, 15), (9, 16)
They don't need to be prime!
📐 Divisibility Rules
Divisible by 2
Last digit is even (0, 2, 4, 6, 8)
Example: 24, 156, 3,248 ✓
Divisible by 3
Sum of digits is divisible by 3
Example: 123 → 1+2+3 = 6 → 6÷3 = 2 ✓
Example: 5,421 → 5+4+2+1 = 12 → 12÷3 = 4 ✓
Divisible by 4
Last TWO digits divisible by 4
Example: 2,316 → 16÷4 = 4 ✓
Example: 7,428 → 28÷4 = 7 ✓
Divisible by 5
Last digit is 0 or 5
Example: 125, 340, 1,995 ✓
Divisible by 6
Divisible by BOTH 2 AND 3
Example: 42 → even ✓, sum = 6 ✓
Divisible by 8
Last THREE digits divisible by 8
Example: 5,128 → 128÷8 = 16 ✓
Divisible by 9
Sum of digits is divisible by 9
Example: 729 → 7+2+9 = 18 → 18÷9 = 2 ✓
Divisible by 10
Last digit is 0
Example: 120, 5,670, 100 ✓
Divisible by 11
Difference of (sum of odd position digits) - (sum of even position digits) is 0 or divisible by 11
Example: 1,331
Odd positions (1st, 3rd): 1 + 3 = 4
Even positions (2nd, 4th): 3 + 1 = 4
Difference = 4 - 4 = 0 ✓
Divisible by 12
Divisible by BOTH 3 AND 4
🔑 HCF & LCM
HCF (Highest Common Factor)
Also called GCD (Greatest Common Divisor)
Definition: Largest number that divides all given numbers.
Methods:
1. Prime Factorization Method
Find HCF of 12 and 18:
12 = 2² × 3
18 = 2 × 3²
HCF = 2¹ × 3¹ = 6 (take minimum powers)
2. Division Method (Euclidean Algorithm)
HCF of 48 and 18:
48 = 18 × 2 + 12
18 = 12 × 1 + 6
12 = 6 × 2 + 0
HCF = 6 (last non-zero remainder)
LCM (Lowest Common Multiple)
Definition: Smallest number divisible by all given numbers.
Prime Factorization Method
Find LCM of 12 and 18:
12 = 2² × 3
18 = 2 × 3²
LCM = 2² × 3² = 36 (take maximum powers)
HCF × LCM Formula
For two numbers a and b:
HCF × LCM = a × b
Example: Numbers are 12 and 18
HCF = 6, a × b = 216
LCM = 216 / 6 = 36 ✓
💡 Solved Examples
Example 1: Divisibility Test
Q: Is 5,832 divisible by 8?
Solution:
Last three digits = 832
832 ÷ 8 = 104 (exactly divisible)
Yes, 5,832 is divisible by 8 ✓
Example 2: Finding HCF
Q: Find HCF of 24, 36, and 48.
Solution:
24 = 2³ × 3
36 = 2² × 3²
48 = 2⁴ × 3
HCF = 2² × 3¹ = 4 × 3 = 12
Answer: 12
Example 3: Finding LCM
Q: Find LCM of 12, 15, and 20.
Solution:
12 = 2² × 3
15 = 3 × 5
20 = 2² × 5
LCM = 2² × 3 × 5 = 60
Answer: 60
Example 4: HCF-LCM Problem
Q: HCF of two numbers is 12, LCM is 180. If one number is 36, find the other.
Solution:
HCF × LCM = Product of two numbers
12 × 180 = 36 × Other number
2,160 = 36 × Other number
Other number = 2,160 / 36 = 60
Answer: 60
Example 5: Co-prime Numbers
Q: Are 35 and 48 co-prime?
Solution:
35 = 5 × 7
48 = 2⁴ × 3
No common factors
HCF = 1
Yes, they are co-prime! ✓
Example 6: Largest Number Dividing
Q: Find largest number that divides 245 and 1,029, leaving remainders 5 and 9 respectively.
Solution:
The number divides (245 - 5) and (1,029 - 9)
= Divides 240 and 1,020
Find HCF of 240 and 1,020:
240 = 2⁴ × 3 × 5
1,020 = 2² × 3 × 5 × 17
HCF = 2² × 3 × 5 = 60
Answer: 60
Example 7: Least Number
Q: Find least number which when divided by 12, 15, 20 leaves remainder 5 in each case.
Solution:
Number = LCM(12, 15, 20) + 5
LCM(12, 15, 20) = 60
Number = 60 + 5 = 65
Answer: 65
📊 Important Number Properties
Sum of First n Natural Numbers
Sum = n(n + 1) / 2
Example: Sum of 1 to 100
= 100 × 101 / 2 = 5,050
Sum of First n Even Numbers
Sum = n(n + 1)
Example: Sum of 2, 4, 6, ..., 20 (n = 10)
= 10 × 11 = 110
Sum of First n Odd Numbers
Sum = n²
Example: Sum of 1, 3, 5, ..., 19 (n = 10)
= 10² = 100
Sum of Squares
1² + 2² + 3² + ... + n² = n(n+1)(2n+1) / 6
Example: 1² + 2² + ... + 10²
= 10 × 11 × 21 / 6 = 385
Sum of Cubes
1³ + 2³ + 3³ + ... + n³ = [n(n+1)/2]²
Example: 1³ + 2³ + ... + 5³
= [5 × 6 / 2]² = 15² = 225
🎯 Number Operations
Even/Odd Rules
Even + Even = Even
Odd + Odd = Even
Even + Odd = Odd
Even × Even = Even
Odd × Odd = Odd
Even × Odd = Even
Remainder Rules
If a leaves remainder r₁ when divided by n
If b leaves remainder r₂ when divided by n
Then:
(a + b) leaves remainder (r₁ + r₂) mod n
(a - b) leaves remainder (r₁ - r₂) mod n
(a × b) leaves remainder (r₁ × r₂) mod n
⚡ Quick Shortcuts
Shortcut 1: Checking Divisibility by 11
Alternate sum method:
321 → 3 - 2 + 1 = 2 (not divisible by 11)
121 → 1 - 2 + 1 = 0 (divisible by 11) ✓
Shortcut 2: Finding Number of Factors
If n = p₁^a × p₂^b × p₃^c
Number of factors = (a+1)(b+1)(c+1)
Example: 12 = 2² × 3¹
Factors = (2+1)(1+1) = 6
{1, 2, 3, 4, 6, 12} ✓
Shortcut 3: Sum of Factors
If n = p₁^a × p₂^b
Sum = [(p₁^(a+1) - 1)/(p₁ - 1)] × [(p₂^(b+1) - 1)/(p₂ - 1)]
Example: 6 = 2¹ × 3¹
Sum = [(2² - 1)/1] × [(3² - 1)/2]
= 3 × 4 = 12
Factors: 1 + 2 + 3 + 6 = 12 ✓
Shortcut 4: Unit Digit Patterns
Powers of 2: 2, 4, 8, 6, 2, 4, 8, 6... (cycle of 4)
Powers of 3: 3, 9, 7, 1, 3, 9, 7, 1... (cycle of 4)
Powers of 4: 4, 6, 4, 6... (cycle of 2)
Powers of 5: Always 5
Powers of 6: Always 6
Powers of 7: 7, 9, 3, 1, 7, 9, 3, 1... (cycle of 4)
Powers of 8: 8, 4, 2, 6, 8, 4, 2, 6... (cycle of 4)
Powers of 9: 9, 1, 9, 1... (cycle of 2)
🔢 Special Numbers
Perfect Numbers
Sum of proper divisors = Number itself
6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14
Armstrong Numbers
Sum of cubes of digits = Number
153 = 1³ + 5³ + 3³ = 1 + 125 + 27 = 153 ✓
370 = 3³ + 7³ + 0³ = 27 + 343 + 0 = 370 ✓
Palindrome Numbers
Same when read forwards or backwards
121, 1331, 12321
⚠️ Common Mistakes
❌ Mistake 1: 1 is Prime
Wrong: 1 is a prime number ✗
Right: 1 is neither prime nor composite ✓
❌ Mistake 2: HCF > LCM
Wrong: HCF can be greater than LCM ✗
Right: HCF ≤ smaller number ≤ larger number ≤ LCM ✓
❌ Mistake 3: Divisibility by 6
Wrong: Divisible by 6 = Divisible by 2 OR 3 ✗
Right: Divisible by 6 = Divisible by 2 AND 3 ✓
❌ Mistake 4: Co-prime = Both Prime
Wrong: Co-prime means both numbers are prime ✗
Right: Co-prime means HCF = 1 (can be composite) ✓
Example: 8 and 9 are co-prime but both composite!
📝 Practice Problems
Level 1:
- Is 7,524 divisible by 9?
- Find HCF of 18 and 24.
- Find LCM of 6 and 8.
Level 2:
- Find largest number dividing 70 and 125 leaving remainders 5 and 8.
- HCF = 6, LCM = 72, one number = 24. Find other number.
- Find sum of first 50 natural numbers.
Level 3:
- Find least number which when divided by 6, 8, 12 leaves remainder 3.
- How many factors does 180 have?
- Find unit digit of 7⁴⁵.
🔗 Related Topics
Uses Number System:
- Simplification - BODMAS with numbers
- Number Series - Pattern recognition
- Average - Sum formulas
- Time & Work - LCM method
Practice:
🎯 Continue Your Learning Journey
Master Number System - The foundation of Quantitative Aptitude! 🔢
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