Numbers, Quantification and Numerical Applications

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Modular Arithmetic is a system of arithmetic for integers that deals with remainders only. X mod Y = R means when X is divided by Y, the remainder is R. For example: 29 mod 3 = 2 (since 29 = 3 × 9 + 2

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1. Addition: (A + B) mod C = (A mod C + B mod C) mod C 2. Subtraction: (A - B) mod C = (A mod C - B mod C) mod C 3. Multiplication: (A × B) mod C = (A mod C × B mod C) mod C These properties allow us

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a ≡ b (mod m) means 'a is congruent to b modulo m'. Two conditions must be satisfied: 1. (a - b) is divisible by m 2. a mod m = b mod m For example: 41 ≡ 21 (mod 5) because 41 - 21 = 20 is divisible b

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Euler's Totient Function φ(n) counts the number of positive integers ≤ n that are coprime to n (gcd = 1 with n). For φ(12): Since 12 = 2² × 3 Using the formula: φ(n) = n(1 - 1/p₁)(1 - 1/p₂)... φ(12)

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τ(n) = number of positive divisors of n σ(n) = sum of all positive divisors of n For n = 12: Positive divisors of 12: 1, 2, 3, 4, 6, 12 τ(12) = 6 (count of divisors) σ(12) = 1 + 2 + 3 + 4 + 6 + 12 =

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μ(n) = { 0, if n has repeated prime factors 1, if n = 1 (-1)^k, if n is product of k distinct primes } μ(12): Since 12 = 2² × 3 (has repeated prime factor 2²), μ(12) = 0 μ(35): Since 35 = 5 × 7 (prod

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Alligation helps find the ratio in which ingredients of different costs should be mixed to produce a mixture of desired mean price. Rule: The ratio of quantities is inversely proportional to differen

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Remaining original liquid = x(1 - y/x)ⁿ liters Where: - x = total capacity of container - y = amount removed each time - n = number of times operation is repeated Example: 40L milk, remove 4L and re