Find the unique prime factors of any integer greater than 1.
Prime Factors:
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What is Prime Decomposition?
Prime decomposition, also known as prime factorization, is the process of breaking down a composite number into its prime factors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11). A composite number is any integer greater than 1 that is not prime.
The Fundamental Theorem of Arithmetic states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers, and that this representation is unique, apart from the order of the factors. For example, the prime decomposition of 12 is 2 x 2 x 3, or 22 x 3.
How it Works (The Algorithm)
To find the prime factors of a number N, we can use the following iterative approach:
Start with the smallest prime number, 2.
While N is divisible by 2, add 2 to our list of factors and divide N by 2.
Move to the next prime number, 3.
While N is divisible by 3, add 3 to our list of factors and divide N by 3.
Continue this process with subsequent odd numbers (5, 7, 11, etc.). We only need to check for divisibility up to the square root of the original number. If, after checking all primes up to the square root, N is still greater than 1, then the remaining value of N is itself a prime factor.
Use Cases for Prime Decomposition:
Simplifying Fractions: Finding the greatest common divisor (GCD) of the numerator and denominator.
Number Theory Research: Fundamental for many advanced concepts in mathematics.
Cryptography: Algorithms like RSA rely on the difficulty of factoring large numbers into their prime components.
Educational Tool: Helping students understand the building blocks of numbers.
function isPrime(num) {
if (num <= 1) return false;
if (num <= 3) return true;
if (num % 2 === 0 || num % 3 === 0) return false;
for (var i = 5; i * i <= num; i = i + 6) {
if (num % i === 0 || num % (i + 2) === 0) return false;
}
return true;
}
function calculatePrimeDecomposition() {
var numberInput = document.getElementById("numberToDecompose");
var primeFactorsDisplay = document.getElementById("primeFactorsDisplay");
var number = parseInt(numberInput.value);
if (isNaN(number) || number = 2) {
if (tempNumber % divisor === 0) {
factors.push(divisor);
tempNumber /= divisor;
} else {
divisor++;
// Optimization: if current divisor squared is greater than remaining number,
// the remaining number must be prime (or 1 if it was perfectly divisible).
if (divisor * divisor > tempNumber && tempNumber > 1) {
factors.push(tempNumber);
break;
}
// Skip even numbers after 2
if (divisor > 2 && divisor % 2 === 0) {
divisor++;
}
}
}
if (factors.length === 0 && isPrime(number)) {
factors.push(number); // If the number itself is prime
}
if (factors.length === 0 && number >= 2) { // Case where number itself is prime
factors.push(number);
}
if (factors.length > 0) {
primeFactorsDisplay.textContent = factors.join(' × ');
primeFactorsDisplay.style.color = "#28a745"; /* Green for success */
} else if (number >= 2) {
// This case should ideally not be reached if logic is sound, but as a fallback
primeFactorsDisplay.textContent = "Could not decompose.";
primeFactorsDisplay.style.color = "#dc3545";
}
}