Real Numbers

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Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. For example: 12 = 2² × 3 (unique f

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For any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. Here, a is the dividend, b is the divisor, q is the quotient, and r is the remainder.

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A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, etc. Note: 1 is not considered a prime number.

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A composite number is a positive integer greater than 1 that has more than two factors. In other words, it can be expressed as a product of two or more prime numbers. Examples: 4, 6, 8, 9, 10, 12, 14,

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60 = 2² × 3 × 5 Step-by-step: 60 ÷ 2 = 30 30 ÷ 2 = 15 15 ÷ 3 = 5 5 ÷ 5 = 1 Therefore, 60 = 2² × 3 × 5

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HCF is the product of the lowest powers of all common prime factors. Example: For 18 = 2 × 3² and 24 = 2³ × 3, common factors are 2 and 3. HCF = 2¹ × 3¹ = 6 (taking lowest powers).

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LCM is the product of the highest powers of all prime factors (common and uncommon). Example: For 18 = 2 × 3² and 24 = 2³ × 3, LCM = 2³ × 3² = 8 × 9 = 72 (taking highest powers).

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For any two positive integers a and b: HCF(a,b) × LCM(a,b) = a × b This relationship is very useful for finding one when the other is known.