Numbers and Quantification

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A prime number is a positive integer p > 1 that has exactly two positive divisors: 1 and p itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23. Note: 2 is the only even prime number.

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Every positive integer n > 1 can be written as a product of primes, and this representation is unique up to the order of primes. Form: n = p₁^a₁ × p₂^a₂ × ... × pᵣ^aᵣ where p₁ < p₂ < ... < pᵣ are dist

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If 1 were considered prime, it would violate the uniqueness part of the Fundamental Theorem of Arithmetic. For example, 10 = 2 × 5 = 1 × 2 × 5 = 1² × 2 × 5, giving multiple factorizations. Since 1 has

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A method to find all prime numbers up to n: 1) List all integers from 2 to n 2) Start with 2, mark it as prime and cross out all its multiples 3) Find next unmarked number, mark as prime and cross out

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RSA is a cryptographic system that uses two large prime numbers p and q to create public key (n=pq, e) and private key (d). Security depends on the difficulty of factoring n into p and q. Large primes

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23 = 16 + 4 + 2 + 1 = 2⁴ + 2² + 2¹ + 2⁰. Therefore, 23 in binary is (10111)₂. Method: Divide by 2 repeatedly: 23÷2=11 R1, 11÷2=5 R1, 5÷2=2 R1, 2÷2=1 R0, 1÷2=0 R1. Read remainders from bottom: 10111.

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(1101)₂ = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8 + 4 + 0 + 1 = 13. General method: Starting from rightmost digit, multiply each digit by corresponding power of 2 and sum all terms.

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i is defined as √(-1), so i² = -1. Powers of i follow a cycle: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, then the pattern repeats. General rule: i⁴ⁿ = 1, i⁴ⁿ⁺¹ = i, i⁴ⁿ⁺² = -1, i⁴ⁿ⁺³ = -i for any integer n ≥