What is a factor and how do you find all factors of a number?
A factor of n is any whole number that divides n with no remainder. To find all factors: test every integer from 1 to √n. If i divides n evenly, both i and n÷i are factors. Example — factors of 36: test 1 (→1,36), 2 (→2,18), 3 (→3,12), 4 (→4,9), 5 (no), 6 (→6,6), stop at √36=6. Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36. A number with exactly two factors (1 and itself) is prime; more than two factors means it's composite.
What is prime factorization and how do you find it?
Prime factorization breaks a number down into a product of prime numbers — the irreducible "atoms" of multiplication. Method: divide by the smallest prime (2) repeatedly, then try 3, 5, 7, 11… Example: 360 = 2 × 180 = 2 × 2 × 90 = 2 × 2 × 2 × 45 = 2³ × 45 = 2³ × 3² × 5. Written in exponential form: 360 = 2³ × 3² × 5. By the Fundamental Theorem of Arithmetic, every integer > 1 has exactly one prime factorization (ignoring order). This is foundational to cryptography (RSA encryption relies on how hard it is to factorise large numbers).
How do you find the Greatest Common Factor (GCF)?
The GCF (also called GCD — Greatest Common Divisor) is the largest number that divides both integers evenly. Method 1 — list factors: factors of 48: {1,2,3,4,6,8,12,16,24,48}; factors of 36: {1,2,3,4,6,9,12,18,36}; GCF = 12. Method 2 — prime factorization: 48 = 2⁴ × 3; 36 = 2² × 3²; GCF = 2² × 3 = 12 (take lowest power of shared primes). Method 3 — Euclidean algorithm: GCF(48,36) = GCF(36,12) = GCF(12,0) = 12 (repeatedly replace larger with remainder). Euclidean is fastest for large numbers.
How do you find the Least Common Multiple (LCM) and when is it used?
LCM is the smallest positive number divisible by both integers. Method 1 — using GCF: LCM(a,b) = (a × b) ÷ GCF(a,b). Example: LCM(48,36) = (48×36) ÷ 12 = 1728 ÷ 12 = 144. Method 2 — prime factorization: 48 = 2⁴ × 3; 36 = 2² × 3²; LCM = 2⁴ × 3² = 144 (take highest power of all primes). Uses: adding fractions (find LCM of denominators: 1/4 + 1/6 = 3/12 + 2/12); scheduling (two events repeating every 48 and 36 days coincide every 144 days).
What is the difference between a prime and a composite number?
A prime number has exactly two factors: 1 and itself (e.g. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29). A composite number has three or more factors (e.g. 4, 6, 8, 9, 10, 12). The number 1 is neither prime nor composite by definition. There are infinitely many primes (proved by Euclid ~300 BCE). The largest known prime (as of 2024) has over 41 million digits. Primes are critical to internet security: RSA encryption uses the product of two large primes as the public key, relying on the computational difficulty of factoring it back.