Number System Formula
Number System formulas and a topic-wise mock test for quantitative aptitude placement exams like TCS NQT, Infosys, and Wipro. Revise, then practice.
Key Number System Formulas
Divisibility Rules - Quick Mental Math
Divisibility rules are shortcuts that help you quickly determine if one number is divisible by another without performing long division.
- Divisible by 2: Last digit must be even (0, 2, 4, 6, 8)
- Divisible by 3: Sum of all digits must be divisible by 3
- Divisible by 4: Last two digits must form a number divisible by 4
- Divisible by 5: Last digit must be 0 or 5
- Divisible by 6: Must be divisible by both 2 AND 3
- Divisible by 8: Last three digits must form a number divisible by 8
- Divisible by 9: Sum of all digits must be divisible by 9
- Divisible by 10: Must end with 0
HCF (GCD) & LCM - Fundamental Concepts
Highest Common Factor (HCF) and Least Common Multiple (LCM) are fundamental concepts in number theory. HCF is the largest number that divides two or more numbers exactly, while LCM is the smallest number that is a multiple of two or more numbers.
- HCF by Prime Factorization: Break numbers into prime factors, take common factors with lowest powers
- Example: HCF of 24 and 36: 24=2³×3, 36=2²×3² → HCF=2²×3=12
- LCM by Prime Factorization: Take all prime factors with highest powers from all numbers
- Example: LCM of 24 and 36: 24=2³×3, 36=2²×3² → LCM=2³×3²=72
- Important Relationship: HCF(a,b) × LCM(a,b) = a × b (for two numbers)
- For three numbers: LCM(a,b,c) = LCM(LCM(a,b), c)
Remainder Theorems - Advanced Number Theory
Remainder theorems provide powerful tools for solving complex problems involving remainders and divisibility without performing actual division.
- Euler's Theorem: If 'a' and 'n' are coprime, then a^φ(n) ≡ 1 (mod n)
- Fermat's Little Theorem: If 'p' is prime and gcd(a,p)=1, then a^(p-1) ≡ 1 (mod p)
- Chinese Remainder Theorem: Solves simultaneous congruences efficiently
- Wilson's Theorem: For prime 'p', (p-1)! ≡ -1 (mod p)
- Example: 7^100 mod 13 → by Fermat 7^12≡1(mod13) → 7^100=7^(8×12+4) ≡ 7^4 ≡ 9 (mod 13)
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