Example 1
\(60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5\)
Prime Factorization
Factor numbers into prime components with detailed step-by-step breakdown
Results
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Prime Factorization
Prime factorization writes a whole number as a product of prime numbers. This representation is unique for every integer greater than 1, apart from the order of the factors.
Key concepts
- Prime number: a number greater than 1 with exactly two positive divisors, 1 and itself.
- Composite number: a number greater than 1 that has factors other than 1 and itself.
- Fundamental theorem of arithmetic: every integer greater than 1 has a unique prime factorization.
- Factor tree method: repeatedly split composite numbers into factor pairs until all leaves are prime.
Applications
- Finding greatest common divisors and least common multiples.
- Simplifying fractions by cancelling common factors.
- Understanding why large prime factors matter in cryptography.
- Solving number theory problems involving divisibility and factors.
Method
- Start with the number you want to factor.
- Divide by the smallest prime factor that works.
- Repeat with the remaining quotient until it is prime.
- Write the product of all prime factors, combining repeated factors with exponents.
Worked Examples
Example 2
\(144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^4 \times 3^2\)
Example 3
\(17 \text{ is prime, so } 17 = 17\)