Prime Factorization Calculator

The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime number itself or can be represented as a unique product of prime numbers. To calculate this, the algorithm starts by dividing the target number (N) by the smallest possible prime number, typically 2. If it divides evenly, the quotient becomes the new target, and the process repeats. If it does not divide evenly, the algorithm tests the next sequential prime (3, 5, 7, 11, etc.) until N is fully reduced to 1. The sequence of all divisors used represents the prime factors.

Let us find the prime factorization of the number 84 using sequential trial division. First, we test the smallest prime number, which is 2. Since 84 is even, 84 ÷ 2 = 42. We record our first prime factor as 2. Next, we take 42 and divide it by 2 again, yielding 21. We record our second prime factor as 2. Now, 21 is no longer divisible by 2, so we test the next prime, which is 3. Since 21 ÷ 3 = 7, we record 3 as our third factor. Finally, 7 is a prime number, so dividing it by itself leaves 1. Compiling our divisors gives us the complete prime factorization: 2 × 2 × 3 × 7, or written in exponential notation, 2² × 3 × 7.

Mastering divisibility rules can greatly accelerate manual factorization checks. For instance, any number ending in an even digit is divisible by 2, while numbers whose digits sum to a multiple of 3 are divisible by 3. When handling very large integers, utilize exponential notation to keep your output organized and easy to read. Always verify your final prime factors by multiplying them back together to ensure they product-match your original input.