Example 1
\(60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5\)
Primfaktorzerlegungs-Rechner
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Prime Factorization
Prime factorization is the process of breaking down a composite number into its prime factors. Every integer greater than 1 is either prime or can be uniquely expressed as a product of primes.
Key Concepts:
- Prime Number: A number greater than 1 with exactly two factors: 1 and itself
- Composite Number: A number with more than two factors
- Fundamental Theorem of Arithmetic: Every integer > 1 has a unique prime factorization
- Factor Tree Method: Repeatedly divide by smallest prime factors
Applications:
- Finding GCD and LCM of numbers
- Simplifying fractions and radicals
- Cryptography (RSA encryption)
- Number theory and mathematics research
Method:
- Divide the number by the smallest prime (2, 3, 5, 7, ...)
- Continue dividing the result by primes
- Stop when you reach 1
- List all prime divisors used
Worked Examples
Example 2
\(144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^4 \times 3^2\)
Example 3
\(17 \text{ is prime, so } 17 = 17\)