Divisibility, Prime Numbers, & Prime Factorization | Math Basics

Factoring is the fundamental mechanism behind simplifying fractions, whether numerical or algebraic. When simplifying 30/50 to 3/5, both numerator and denominator are factored by dividing out common factors (in this case, 10). The same principle applies to algebraic fractions: factor both numerator and denominator completely, then cancel any common factors. For example, (4x² - 1)/(8x - 4) factors to [(2x + 1)(2x - 1)]/[4(2x - 1)], allowing cancellation of the common (2x - 1) factor, leaving (2x + 1)/4. To verify factoring correctness, always multiply the factors back together to confirm you obtain the original expression. This systematic approach—complete factoring followed by careful cancellation—is essential for simplifying complex algebraic expressions.

Prime factorization is a method for finding the lowest common denominator (LCD) of fractions by breaking each denominator into its prime factors and taking the highest power of each prime that appears in any factorization. For example, to find the LCD of 24 (2³ × 3) and 60 (2² × 3 × 5), you take the highest powers: 2³ × 3 × 5 = 120. This LCD is then used to convert fractions to equivalent fractions with the same denominator for addition or subtraction.

To find the lowest common denominator (LCD) of fractions, create a factor tree for each denominator to identify their prime factors, then multiply together the highest power of each prime factor appearing in any denominator; this method works for any pair of fractions and avoids unnecessarily large common denominators.

Prime factorization is the key method for finding the least common denominator (LCD) of two or more fractions. This technique involves breaking down each denominator into its basic building blocks of prime numbers. By expressing denominators as products of prime factors, we can identify the smallest common denominator needed to add or subtract fractions with different denominators.

The LCD of fractions is found by prime factorizing each denominator, identifying all unique prime factors, and multiplying together the highest powers of each prime factor across all denominators; for example, the LCD of 1/25, 3/49, 7/125, 5/64, and 1/7 is calculated as follows: 25 = 5², 49 = 7², 125 = 5³, 64 = 2⁶, and 7 = 7¹, so the LCD = 2⁶ × 5³ × 7² = 64 × 125 × 49 = 392,000.

The Division Algorithm can be written as B ≡ R (mod A), meaning B and R have the same remainder when divided by A. This is equivalent to B - R = A × Q. The Euclidean Algorithm finds the greatest common divisor (GCD) by repeatedly applying the Division Algorithm: for A > B > 0, write A = B × Q + R with 0 ≤ R < B, then gcd(A, B) = gcd(B, R). Repeat until R = 0.

To prove that a number divides another, one can use the Euclidean algorithm. If d divides u_n + 1 and d divides u_n, then d divides their difference (u_n + 1) - u_n = 1. Since the only positive divisors of 1 are 1, d must be 1. This demonstrates how the Euclidean algorithm can be used to prove divisibility properties by showing that the greatest common divisor must be 1.

The Euclidean algorithm is an alternative method for finding LCM and GCD. For LCM: divide all numbers by their smallest common prime factor, repeat until no common factor exists, then multiply all divisors and the last row. For GCD: multiply only the divisors. For 8, 12, 24: LCM = 432; GCD = 12. For 9, 12, 18: LCM = 432; GCD = 6. A composite number is formed by multiplying prime numbers (2, 3, 5, etc.). Divisibility rules: For 2, check if the last digit is even. For 3, check if the sum of digits is divisible by 3. For 5, check if the last digit is 0 or 5.

The Euclidean division algorithm states that for any natural numbers a and b (b ≠ 0), there exist unique integers q (quotient) and r (remainder) such that a = b × q + r, where 0 ≤ r < b. The remainder is always less than the divisor and non-negative. For example, 2900 ÷ 13 = 225 with remainder 1, expressed as 2900 = 13 × 225 + 1. Similarly, 1715 ÷ 13 = 131 with remainder 12, expressed as 1715 = 13 × 131 + 12. In modular arithmetic, if a ≡ b (mod n), then a + k ≡ b + k (mod n) for any integer k. This means adding the same value to both sides preserves the congruence. For example, if b ≡ 12 (mod 13), then b + 1 ≡ 13 ≡ 0 (mod 13). A number is divisible by n if and only if its remainder modulo n equals 0. When solving problems involving divisibility, we must find the remainder and verify it equals 0.

This comprehensive lesson covers fundamental concepts in number theory: (1) Divisibility: b is divisible by a (a ≠ 0) if b = a × k for some integer k, denoted a | b; non-divisibility (a ∤ b) occurs when a remainder exists; (2) Properties: a | 0 for any a ≠ 0; a | 1 iff a = ±1; if a|b and a|c, then a|(b+c) and a|(b-c); if a|b and b|c, then a|c; if a|b and b ≠ 0, then |a| ≤ |b|; (3) Greatest Common Divisor (GCD): For integers a and b (not both zero), gcd(a, b) is the largest positive integer dividing both, satisfying: d|a and d|b, and if c|a and c|b, then c|d; (4) Bézout's Identity: gcd(a, b) = ax + by for some integers x, y; (5) Euclidean Algorithm: Repeatedly apply division: a = qb + r1, b = q2r1 + r2, r1 = q3r2 + r3, until remainder is zero; the last non-zero remainder is the GCD.

Prime factorization has important real-world applications in cryptography, particularly in RSA encryption. This method, used for secure online transactions (like credit card payments), relies on the difficulty of factoring large composite numbers into their prime factors. The security of RSA depends on the fact that while multiplying primes is easy, reversing this process (factoring) is computationally difficult for large numbers.

A prime number is an integer > 1 divisible only by 1 and itself (e.g., 2, 3, 5, 7, 1997). Prime testing is computationally efficient, but prime factorization of large numbers is computationally difficult. This asymmetry (easy to multiply, hard to factor) forms the basis of RSA security. Any composite number can be uniquely expressed as a product of primes.

RSA cryptography, developed in the 1970s, secures internet communications by relying on the computational difficulty of factoring large numbers into their prime components. The system uses two large prime numbers to generate a public key for encryption and a private key for decryption. While multiplying primes is easy, reversing this process (factoring) is extremely difficult even for powerful computers, ensuring data security.

RSA cryptography, which secures modern electronic transactions, relies on the difficulty of factoring large composite numbers into their prime factors. This is based on the Fundamental Theorem of Arithmetic. The security of RSA depends on the unknown distribution of prime numbers, which is why finding the prime factors of large numbers is computationally difficult.