Number System - Complete Guide for SSC CGL
What is Number System? It's the foundation of mathematics that deals with different types of numbers and their properties. For SSC CGL, mastering Number System can help you solve 4-6 questions quickly with high accuracy.
Pro Tip – Number System is Scoring!
This topic appears in every SSC CGL exam. With proper understanding and practice, you can score full marks. Visit SKY Practice for 500+ practice questions with video solutions.
1. Divisibility Rules (SSC Shortcuts)
What is Divisibility? A number is divisible by another if it can be divided exactly without leaving a remainder.
Must-Know Divisibility Rules
Understanding Divisibility
Divisibility rules help you quickly check if one number divides another without actual division. These rules are based on the properties of numbers in decimal system.
Divisible by 2
Rule: Last digit is even (0,2,4,6,8)
✓ Yes, 5,684 is divisible by 2
Divisible by 3
Rule: Sum of all digits divisible by 3
12 ÷ 3 = 4 (exact)
✓ Yes, 1,236 is divisible by 3
Divisible by 4
Rule: Last two digits divisible by 4
16 ÷ 4 = 4 (exact)
✓ Yes, 7,316 is divisible by 4
Divisible by 5
Rule: Last digit 0 or 5
✓ Yes, 785 is divisible by 5
Divisible by 6
Rule: Divisible by both 2 AND 3
Step 2: Sum = 1+3+8 = 12 (divisible by 3) ✓
✓ Yes, 138 is divisible by 6
Divisible by 8
Rule: Last three digits divisible by 8
128 ÷ 8 = 16 (exact)
✓ Yes, 45,128 is divisible by 8
SSC Shortcut: Divisibility by 7
Method: Double the last digit → Subtract from remaining number → Repeat
Example: Check 672 ÷ 7
Step 1: Last digit = 2, double = 4
Step 2: Remaining number = 67, 67 - 4 = 63
Step 3: 63 ÷ 7 = 9 (exact)
✓ 672 is divisible by 7
Example: Check 1,4 6,4 1 (odd places: 1,6,1; even places: 4,4)
Odd sum = 1+6+1 = 8
Even sum = 4+4 = 8
Difference = 8-8 = 0 ✓ Divisible by 11
2. HCF & LCM Formulas & Tricks
HCF (Highest Common Factor): Largest number that divides all given numbers.
LCM (Least Common Multiple): Smallest number divisible by all given numbers.
Calculation Methods
How to Find HCF
- Factorization Method: List all factors, find common ones
- Prime Factorization: Common primes with lowest power
- Division Method: Repeated division till remainder zero
- Formula: HCF × LCM = Product of two numbers
Common Mistakes
- Mixing up HCF and LCM concepts
- Using wrong formula for more than 2 numbers
- Forgetting 1 is always a factor
- Not checking prime factors properly
SSC Shortcut: HCF of Fractions
Formula: HCF of Numerators ÷ LCM of Denominators
Example: Find HCF of 2/3, 4/5, 6/7
Step 1: HCF of numerators (2,4,6) = 2
Step 2: LCM of denominators (3,5,7) = 105
Step 3: HCF = 2 ÷ 105 = 2/105
SSC Shortcut: LCM of Fractions
Formula: LCM of Numerators ÷ HCF of Denominators
Example: Find LCM of 2/3, 4/5, 6/7
Step 1: LCM of numerators (2,4,6) = 12
Step 2: HCF of denominators (3,5,7) = 1
Step 3: LCM = 12 ÷ 1 = 12
Solved Example: Real SSC Question
Step-by-step solution:
1. Let numbers = 4x and 5x
2. HCF of (4x, 5x) = x (since 4 and 5 are co-prime)
3. Given HCF = 12, so x = 12
4. Numbers = 4×12 = 48 and 5×12 = 60
5. LCM(48, 60) = ?
Prime factors: 48 = 2⁴×3, 60 = 2²×3×5
LCM = 2⁴ × 3 × 5 = 16 × 3 × 5 = 240
Final Answer: 240
Time taken: 30 seconds with practice
3. Remainder Theorem & Modular Arithmetic
What is Remainder? When we divide a number, the leftover part is remainder.
Important Rules & Concepts
Understanding Remainders
Remainder theorem helps find remainders without actual division. Modular arithmetic (a mod b) means remainder when a is divided by b.
Basic Rules
- (a+b) mod m = [(a mod m)+(b mod m)] mod m
- (a×b) mod m = [(a mod m)×(b mod m)] mod m
- aⁿ mod m = (a mod m)ⁿ mod m
- Negative remainders: Add divisor to make positive
Euler's Theorem (Advanced)
- If HCF(a, n) = 1, then a^φ(n) ≡ 1 (mod n)
- φ(n) = Euler's totient function
- For prime p: φ(p) = p-1
- Useful for large powers
Chinese Remainder Theorem
- Solve simultaneous congruences
- Find x such that:
x ≡ a (mod m)
x ≡ b (mod n) - Important for number puzzles
SSC Trick: Remainder by Binomial Theorem
Method: Express number as (multiple of divisor + remainder)
Example: Find remainder when 17²³ ÷ 16
17 = 16 + 1
17²³ = (16 + 1)²³
= 16²³ + ²³C₁16²² + ... + ²³C₂₂16 + ¹
All terms except last have 16 as factor
Remainder = 1
Solved Example: SSC Pattern
Odd³ = odd, Even³ = even multiple of 8
Step 2: 2³ = 8 (divisible by 4, remainder 0)
4³, 6³, 8³,... all divisible by 4
Step 3: Only odd cubes matter
1³ = 1 → remainder 1
3³ = 27 → remainder 3
5³ = 125 → remainder 1
7³ = 343 → remainder 3
Pattern: 1,3,1,3,...
Step 4: From 1 to 99, there are 50 odd numbers
Pairs of (1,3) sum to 4 (divisible by 4)
Final Answer: Remainder = 0
Key: Pattern identification saves time
4. Unit Digit & Last Two Digits
Why important? Questions like "Find unit digit of 7⁹⁵ × 8⁶⁴" appear frequently.
Unit Digit Cycles
Understanding Cycles
Unit digits repeat in cycles. For example, powers of 2: 2¹=2, 2²=4, 2³=8, 2⁴=16(6), 2⁵=32(2) → cycle of 4.
1
Digits 0, 1, 5, 6
Always same unit digit:
- 0ⁿ → 0 (for n≥1)
- 1ⁿ → 1
- 5ⁿ → 5
- 6ⁿ → 6
Example: 12345⁵⁶ → unit digit = 5
2
Digits 4, 9
Cycle length = 2:
- 4¹→4, 4²→6, 4³→4,...
- 9¹→9, 9²→1, 9³→9,...
Rule: Check if power is odd or even
Example: 4⁹⁹ → 99 odd → unit digit = 4
4
Digits 2, 3, 7, 8
Cycle length = 4:
- 2→4→8→6→2...
- 3→9→7→1→3...
- 7→9→3→1→7...
- 8→4→2→6→8...
Divide power by 4, use remainder
SSC 4-Step Method
- Step 1: Find unit digit of base
- Step 2: Determine cycle length
- Step 3: Divide power by cycle length
- Step 4: Use remainder to find position
Solved Example: SSC Question
Cycle: 7→9→3→1 (length 4)
95 ÷ 4 = 23 remainder 3
3rd position in cycle = 3
For 8⁶⁴:
Cycle: 8→4→2→6 (length 4)
64 ÷ 4 = 16 remainder 0
Remainder 0 means last position = 6
Final: 3 × 6 = 18 → unit digit = 8
Answer: 8 (Time: 45 seconds)
5. Prime Numbers & Integer Properties
Prime Numbers: Numbers greater than 1 with only two factors: 1 and itself.
Essential Facts & Rules
Must Remember Facts
- Smallest prime = 2 (only even prime)
- 1 is NOT prime (nor composite)
- All primes except 2 & 3 are of form 6k±1
- There are infinite primes (proved by Euclid)
- Twin primes: (3,5), (5,7), (11,13), etc.
Common Confusions
- 1 is neither prime nor composite
- 2 is prime (many forget it's even)
- 0 is not considered for prime/composite
- Negative numbers not in prime discussion
SSC Trick: Prime Checking
Check divisibility by primes ≤ √n
Example: Is 167 prime?
√167 ≈ 12.9
Check primes ≤ 12: 2,3,5,7,11
167 not divisible by any → Prime ✓
Total divisors = (a+1)(b+1)(c+1)...
Example: Find divisors of 72
72 = 2³ × 3²
Total divisors = (3+1)(2+1) = 4×3 = 12
Divisors: 1,2,3,4,6,8,9,12,18,24,36,72
Ready to Master Number System?
Access 500+ Number System questions with detailed solutions, video explanations, and chapter-wise tests
Start Number System PracticeIncludes divisibility, HCF/LCM, remainder theorem, unit digit, and prime numbers
Frequently Asked Questions
Q1: How many questions from Number System in SSC CGL?
Answer: Typically 4-6 questions appear from Number System in Tier-I. In Tier-II (Quantitative Abilities), you can expect 8-12 questions. It's one of the highest weightage topics in Quant.
Q2: What's the fastest way to check divisibility by 7?
Answer: Use this shortcut: Double the last digit, subtract from the remaining number. Repeat until you get a manageable number. Example: 672 → 67-4=63 → 63÷7=9 ✓.
Q3: How to find unit digit of large powers quickly?
Answer: Follow the 4-step method: 1) Find unit digit of base, 2) Determine cycle length (1,2, or 4), 3) Divide power by cycle length, 4) Use remainder to find position in cycle.
Q4: Is 1 a prime number?
Answer: No. 1 is neither prime nor composite. By definition, prime numbers have exactly two distinct factors: 1 and itself. 1 has only one factor (itself), so it doesn't qualify.
Q5: What's the difference between HCF and LCM?
Answer: HCF (Highest Common Factor) is the largest number dividing all given numbers. LCM (Least Common Multiple) is the smallest number divisible by all given numbers. Remember: HCF × LCM = Product of two numbers.
Q6: How to solve remainder problems quickly?
Answer: Use modular arithmetic rules: (a+b) mod m = [(a mod m)+(b mod m)] mod m. For large powers, use remainder cycles or binomial expansion. Practice with patterns like 2,4,8,6 cycle for powers of 2.
Final Exam Strategy for Number System
Time Allocation: Spend 30-40 seconds per Number System question.
Priority Order: 1) Divisibility (quickest), 2) Unit digit, 3) HCF/LCM, 4) Remainder, 5) Prime numbers.
Accuracy Check: Always verify with small values if unsure.
👉 For complete mastery with 1000+ questions, visit SKY Practice!