Number system

1. Types of numbers and their properties

Natural numbers are the counting numbers 1, 2, 3 and so on; whole numbers additionally include 0. Some mathematical conventions include 0 among natural numbers, so follow any definition supplied in the question. Integers include negative whole numbers, zero and positive whole numbers. Rational numbers can be expressed as p/q, where p and q are integers and q is non-zero. Irrational numbers, such as √2 and π, cannot be expressed in this form. Rational and irrational numbers together constitute the real numbers.

A rational number has a decimal expansion that either terminates or eventually repeats. An irrational number has a non-terminating, non-repeating decimal expansion. After a rational fraction is reduced to its lowest terms, its decimal terminates exactly when the denominator contains no prime factors other than 2 and 5. Thus, 7/40 terminates, whereas 7/30 repeats. The reduction condition is essential: 3/6 terminates because it simplifies to 1/2.

An integer is even if divisible by 2 and odd otherwise; zero is even. Even plus even and odd plus odd are even, while even plus odd is odd. A product is odd only when every integer factor is odd. These parity rules can eliminate options without finding exact values. However, do not transfer integer properties to arbitrary fractions or decimals.

• Zero is neither positive nor negative. Division by zero is undefined.
• The sum or product of two rational numbers is rational.
• Two irrational numbers may have a rational sum or product: √2 + (−√2) = 0 and √2 × √2 = 2.
• The square of every integer is non-negative; the principal square root symbol denotes the non-negative root.

2. Divisibility, primes and prime factorisation

A prime number is a positive integer greater than 1 with exactly two positive factors: 1 and itself. A composite number has more than two positive factors. The number 1 is neither prime nor composite; 2 is the only even prime. To test whether an integer n greater than 1 is prime, check divisibility only by primes not exceeding √n. Any composite number must have at least one prime factor within that limit.

Divisibility tests reduce calculation. A number is divisible by 2 if its units digit is even, by 5 if it ends in 0 or 5, and by 10 if it ends in 0. Divisibility by 3 or 9 depends on the digit sum. For 4 and 8, examine the last two and last three digits respectively. For 11, the difference between the sums of digits in alternating positions must be zero or a multiple of 11.

Prime factorisation converts many questions into comparisons of exponents. For example, 360 = 2³ × 3² × 5. A perfect square has even exponents for all prime factors; a perfect cube has exponents divisible by 3. Therefore, the smallest positive integer by which 360 must be multiplied to obtain a perfect square is 10: this makes the exponents of both 2 and 5 even.

• To test divisibility by 6, check both 2 and 3; for 12, check both 3 and 4.
• Divisibility by a and b guarantees divisibility by ab only when a and b are coprime.
• Coprime numbers have HCF 1 but need not be prime: 8 and 15 are coprime.

3. Factors, HCF and LCM

The highest common factor, also called the greatest common divisor, is the largest positive integer dividing each of the given integers. The least common multiple is their smallest common positive multiple. In prime factorisation, the HCF takes the minimum exponent of each prime across the numbers, treating an absent prime as having exponent zero. The LCM takes the maximum exponent. For 72 = 2³ × 3² and 120 = 2³ × 3 × 5, the HCF is 24 and the LCM is 360.

Use HCF when dividing quantities into the largest equal-sized units without any remainder. Use LCM when finding the earliest recurrence of events with fixed integer periods, assuming they begin together. For two positive integers, their product equals the product of their HCF and LCM. This identity does not generally extend to three or more integers, a frequent trap in objective questions.

If n = pᵃqᵇrᶜ for distinct primes, its number of positive factors is (a + 1)(b + 1)(c + 1), because each prime exponent can be independently selected from zero to its maximum. Thus, 72 has 4 × 3 = 12 positive factors. A positive integer has an odd number of positive factors exactly when it is a perfect square, since all other factors pair with distinct complementary factors.

• Euclid’s algorithm uses repeated division: HCF(252, 105) = HCF(105, 42) = HCF(42, 21) = 21.
• The smallest positive integer divisible by each number in a given set is their LCM.
• Read whether the question asks for positive factors, prime factors or distinct prime factors; these counts differ.

4. Remainders, cyclicity and factorials

The division algorithm writes an integer N as N = dq + r, where d is a positive divisor, q is an integer and 0 ≤ r < d. Numbers with the same remainder on division by d are congruent modulo d. In sums and products, large numbers can be replaced by their remainders before calculation. For example, 47 × 53 leaves the same remainder modulo 5 as 2 × 3, namely 1.

Powers often follow repeating remainder cycles. The units digits of powers of 7 repeat as 7, 9, 3, 1. Since 103 leaves remainder 3 when divided by 4, the units digit of 7¹⁰³ is 3. If the exponent is divisible by the cycle length, use the last member of the cycle, not a nonexistent zeroth member.

A factorial n! is the product of positive integers from 1 to n, with 0! defined as 1. Trailing zeros come from factors of 10, each formed by pairing 2 with 5. In n!, factors of 5 are scarcer, so count them using ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ and further terms until zero. Consequently, 100! has 20 + 4 = 24 trailing zeros.

• If several numbers leave the same remainder on division by d, d divides their pairwise differences.
• For negative integers, retain a non-negative standard remainder: −7 = 5 × (−2) + 3.
• Do not cancel a common factor in a congruence while retaining the modulus unless that factor is coprime to the modulus.

5. Digit problems and an efficient CSAT approach

Place value translates digit puzzles into algebra. A two-digit number with tens digit a and units digit b equals 10a + b, where a ranges from 1 to 9 and b from 0 to 9. Its reversed value is 10b + a. Their difference is 9(a − b), necessarily divisible by 9. Similarly, a three-digit number abc equals 100a + 10b + c; digit symbols are not multiplied together.

Before calculating, identify the domain, constraints and required quantity. Check whether zero, negative integers, repeated digits or leading zeros are allowed. Apply parity, divisibility and size bounds to eliminate impossible options. Use substitution from options when it is faster than solving an equation, but verify every condition. A candidate answer satisfying only one of several remainder conditions is insufficient.

• Write short working notes: divisor, permitted remainder, digit restrictions and whether the least positive value is required.
• Estimate magnitude before exact calculation to detect misplaced zeros or incorrect powers.
• Under negative marking, prioritise questions with a clear method; avoid spending disproportionate time on an unfamiliar pattern.

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