Natural Numbers (N): Counting numbers starting from 1. Examples: 1, 2, 3, 4, 5... They are also called positive integers.
Whole Numbers (W): Natural numbers including 0. Examples: 0, 1, 2, 3, 4... The smallest whole number is 0.
Integers (Z): Include all positive numbers, negative numbers, and zero. Examples:..., -3, -2, -1, 0, 1, 2, 3...
Rational Numbers (Q): Numbers that can be expressed in the form p/q where q ≠ 0 and p, q are integers. Examples: 1/2, -3/4, 0.75, 5.
Irrational Numbers: Numbers that cannot be expressed as p/q. Their decimal expansion is non-terminating and non-repeating. Examples: √2, √3, π, e.
Real Numbers (R): The set of all rational and irrational numbers combined. Every point on the number line represents a real number.
Even Numbers: Integers divisible by 2. Examples:..., -4, -2, 0, 2, 4, 6... Note: 0 is an even number.
Odd Numbers: Integers not divisible by 2. Examples:..., -3, -1, 1, 3, 5, 7...
Co-prime Numbers (Relatively Prime): Two numbers whose HCF is 1. Examples: (8, 15), (7, 9). They need not be prime individually.
Twin Primes: Pair of prime numbers whose difference is 2. Examples: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43).
Perfect Numbers: A number equal to the sum of its proper divisors (excluding itself). Examples: 6 = 1+2+3, 28 = 1+2+4+7+14.
Prime and Composite Numbers
A prime number has exactly two distinct factors: 1 and the number itself.
2 is the smallest and the only even prime number. All other even numbers are composite.
3 is the smallest odd prime number.
1 is neither prime nor composite. It has exactly one factor (itself).
A composite number has more than two factors. The smallest composite number is 4.
All prime numbers greater than 3 can be expressed in the form 6k ± 1 (but not every 6k ± 1 is prime).
To check whether a number N is prime, test its divisibility by all prime numbers up to √N. If none divides it, N is prime.
Example: To check if 97 is prime, √97 ≈ 9.8. Test primes: 2, 3, 5, 7. None divides 97, so 97 is prime.
There are 25 prime numbers between 1 and 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
There are 15 prime numbers between 1 and 50.
Every composite number can be uniquely expressed as a product of prime factors (Fundamental Theorem of Arithmetic).
Goldbach Conjecture: Every even integer greater than 2 can be expressed as the sum of two primes (unproven but verified for very large ranges).
Divisibility Rules
Divisibility by 2: The last digit (units digit) must be 0, 2, 4, 6, or 8.
Divisibility by 3: The sum of all digits must be divisible by 3. Example: 372 → 3+7+2 = 12 → divisible by 3.
Divisibility by 4: The number formed by the last two digits must be divisible by 4. Example: 7324 → 24 ÷ 4 = 6 → divisible.
Divisibility by 5: The last digit must be 0 or 5.
Divisibility by 6: The number must be divisible by both 2 and 3 simultaneously.
Divisibility by 7: Double the last digit and subtract it from the remaining number. If the result is divisible by 7, the original number is also divisible by 7. Example: 203 → 20 - 2×3 = 14 → divisible by 7.
Divisibility by 8: The number formed by the last three digits must be divisible by 8. Example: 17,256 → 256 ÷ 8 = 32 → divisible.
Divisibility by 9: The sum of all digits must be divisible by 9. Example: 729 → 7+2+9 = 18 → divisible by 9.
Divisibility by 10: The last digit must be 0.
Divisibility by 11: The difference between the sum of digits at odd positions and sum of digits at even positions (from right) must be 0 or a multiple of 11. Example: 121 → (1+1) - 2 = 0 → divisible.
Divisibility by 12: The number must be divisible by both 3 and 4.
Divisibility by 13: Multiply the last digit by 4 and add to the remaining number. Repeat until manageable. If result is divisible by 13, the original is too.
Divisibility by 15: The number must be divisible by both 3 and 5.
Divisibility by 25: The last two digits must be divisible by 25 (i.e., 00, 25, 50, or 75).
Factors and Multiples
If a number N = p^a × q^b × r^c (prime factorization), then the total number of factors = (a+1)(b+1)(c+1).
Example: 120 = 2³ × 3¹ × 5¹ → Number of factors = (3+1)(1+1)(1+1) = 4×2×2 = 16.
Sum of all factors of N = [(p^(a+1) - 1)/(p - 1)] × [(q^(b+1) - 1)/(q - 1)] × [(r^(c+1) - 1)/(r - 1)].
Product of all factors of N = N^(total number of factors / 2).
Number of even factors: Remove the contribution of 2^0 from the factorization of 2. If N = 2^a ×..., even factors = a × (other factor counts).
Number of odd factors: Set the power of 2 as 0. If N = 2^a × 3^b × 5^c, odd factors = (b+1)(c+1).
A number is a perfect square if and only if all exponents in its prime factorization are even.
Perfect squares have an odd number of total factors. All other numbers have an even number of factors.
HCF and LCM
HCF (Highest Common Factor) or GCD (Greatest Common Divisor): The largest number that divides all given numbers exactly.
LCM (Least Common Multiple): The smallest number that is exactly divisible by all given numbers.
For two numbers a and b: HCF(a, b) × LCM(a, b) = a × b. This relation holds only for two numbers.
HCF of fractions = HCF of numerators / LCM of denominators.
LCM of fractions = LCM of numerators / HCF of denominators.
HCF can be found using: (1) Prime Factorization Method – take lowest powers of common primes, (2) Division Method (Euclidean Algorithm) – divide larger by smaller repeatedly.
LCM can be found using: (1) Prime Factorization Method – take highest powers of all primes, (2) Common Division Method.
HCF of any two consecutive numbers is always 1. LCM of any two consecutive numbers is their product.
HCF of any two consecutive even numbers is always 2.
If HCF(a, b) = H, then a = Hx and b = Hy where x and y are co-prime.
The largest number that divides a, b, c leaving remainders p, q, r respectively is HCF of (a-p), (b-q), (c-r).
The largest number that divides a, b, c leaving the same remainder is HCF of |a-b|, |b-c|, |a-c|.
The smallest number which when divided by a, b, c leaves no remainder is LCM(a, b, c).
The smallest number which when divided by a, b, c leaves remainders p, q, r respectively (where a-p = b-q = c-r = k): Answer = LCM(a, b, c) - k.
Remainder Theorem and Patterns
Remainder of (a × b) / n = [(Remainder of a/n) × (Remainder of b/n)] mod n.
Remainder of (a + b) / n = [(Remainder of a/n) + (Remainder of b/n)] mod n.
Remainder of (a^n) / d can be found by identifying the cyclicity of remainders.
Fermat Little Theorem: If p is prime and a is not divisible by p, then a^(p-1) ≡ 1 (mod p). Useful for large power remainder problems.
Euler Theorem: a^φ(n) ≡ 1 (mod n) when HCF(a, n) = 1. φ(n) is Euler totient function.
Wilson Theorem: (p-1)! ≡ -1 (mod p) if p is prime. Useful in factorial remainder problems.
When a number is divided by d, the possible remainders are 0, 1, 2,..., (d-1).
If N divided by d gives remainder r, then N = d × q + r where 0 ≤ r < d.
Unit Digit (Cyclicity)
The unit digit of a product depends only on the unit digits of the numbers being multiplied.
Cyclicity of unit digits of powers: 2 → cycle of 4 (2,4,8,6), 3 → cycle of 4 (3,9,7,1), 4 → cycle of 2 (4,6), 5 → always 5, 6 → always 6, 7 → cycle of 4 (7,9,3,1), 8 → cycle of 4 (8,4,2,6), 9 → cycle of 2 (9,1).
Any number ending in 0, 1, 5, or 6 always has the same unit digit regardless of the power.
To find unit digit of a^b: find b mod (cyclicity length) and use the cycle. If remainder is 0, take the last element of the cycle.
Unit digit of n! for n ≥ 5 is always 0 (since 5! = 120 contains factor 10).
Factorials
n! = n × (n-1) × (n-2) ×... × 3 × 2 × 1. By convention, 0! = 1 and 1! = 1.
Highest power of a prime p in n! = ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ +... (Legendre Formula). Example: Highest power of 2 in 10! = 5+2+1 = 8.
Number of trailing zeros in n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ +... (since trailing zeros come from pairs of 2 and 5, and 2s are always more).
Example: Trailing zeros in 100! = 20 + 4 = 24.
Number of trailing zeros in n! in base b: find prime factorization of b and use Legendre formula for the limiting prime factor.
Important Number Properties
Sum of first n natural numbers = n(n+1)/2.
Sum of squares of first n natural numbers = n(n+1)(2n+1)/6.
Sum of cubes of first n natural numbers = [n(n+1)/2]² = (Sum of first n natural numbers)².
Sum of first n even numbers = n(n+1). Sum of first n odd numbers = n².
a² - b² = (a+b)(a-b). This factorization is frequently used in simplification.
a³ + b³ = (a+b)(a² - ab + b²). a³ - b³ = (a-b)(a² + ab + b²).
If a + b + c = 0, then a³ + b³ + c³ = 3abc.
The product of n consecutive integers is always divisible by n!. This holds because among n consecutive integers, the required factors of every integer from 1 to n are present.
Exam Focus & Tricks
Number System questions frequently test divisibility, remainders, HCF, LCM, prime numbers, and unit digits.
To check whether a number is prime, test divisibility by prime numbers up to its square root only.
In remainder-based questions, use cyclicity and modular arithmetic instead of direct large calculations.
LCM is usually indicated by phrases like "when will they meet again", "when will it happen together again", "bells ringing together".
HCF is usually indicated by words like "greatest", "maximum", "largest possible", "equal groups", "largest tile".
For questions asking "largest n-digit number divisible by x": Divide the largest n-digit number by x, subtract the remainder.
For questions asking "smallest n-digit number divisible by x": Divide the smallest n-digit number by x, subtract remainder from x and add to the number.
Remember: Sum of digits divisibility rules (3 and 9) are the fastest tricks in competitive exams.
In "find the number" type questions, use the relationship: Dividend = Divisor × Quotient + Remainder.