Real Numbers

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Division Algorithm: Given positive integers a and b, there exist unique whole numbers q and r such that a = bq + r, where 0 ≤ r < b. Here, a is dividend, b is divisor, q is quotient, and r is remainde…

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Euclid's Algorithm is a technique to find HCF of two given integers. Steps: 1) Apply division algorithm to larger and smaller number 2) If remainder = 0, divisor is HCF 3) If remainder ≠ 0, apply divi…

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Step 1: 100 = 60 × 1 + 40 Step 2: 60 = 40 × 1 + 20 Step 3: 40 = 20 × 2 + 0 Since remainder is 0, HCF(60, 100) = 20…

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

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If p is a prime number and p divides a² (where a is a positive integer), then p also divides a. This is a key theorem used in proving irrationality of numbers like √2.

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A rational number p/q (in lowest terms) has a terminating decimal expansion if and only if the prime factorization of q is of the form 2ⁿ5ᵐ, where n and m are non-negative integers. This is because po…

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A rational number p/q (in lowest terms) has a non-terminating repeating decimal expansion if the prime factorization of q is NOT of the form 2ⁿ5ᵐ. This means q has prime factors other than 2 and 5.

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7/8 = 7/(2³) = (7 × 5³)/(2³ × 5³) = (7 × 125)/(2³ × 5³) = 875/10³ = 875/1000 = 0.875…