Prime Factorization Calculator

Prime factorization is the mathematical process of breaking down a composite number into the exact set of prime numbers that, when multiplied together, result in the original number. This concept serves as the foundational DNA of number theory, underpinning everything from basic fraction simplification in elementary mathematics to the advanced cryptographic systems that secure global digital communications. By mastering prime factorization, you will gain a profound understanding of how numbers are constructed, how they interact, and how to manipulate them for complex mathematical and real-world problem-solving.

What It Is and Why It Matters

To understand prime factorization, you must first view numbers as chemical compounds and prime numbers as the fundamental elements of the periodic table. A prime number is a whole number greater than 1 that cannot be made by multiplying other whole numbers; its only divisors are 1 and itself (for example, 2, 3, 5, 7, 11). A composite number, conversely, is any whole number that can be formed by multiplying two or more smaller whole numbers (for example, 12, which is 3 times 4). Prime factorization is the exact recipe for constructing a specific composite number using only prime numbers. For instance, the prime factorization of 12 is not 3 times 4, because 4 is not a prime number; the correct prime factorization is 2 times 2 times 3.

This concept is not merely an abstract mathematical exercise; it is governed by a bedrock rule called the Fundamental Theorem of Arithmetic. This theorem states that every integer greater than 1 is either a prime number itself or can be represented as a unique product of prime numbers, disregarding the order of the factors. This uniqueness is what makes prime factorization so powerful. If you factor the number 360, you will always arrive at the exact same collection of prime building blocks: three 2s, two 3s, and one 5. Because this prime signature is entirely unique to the number 360, mathematicians and computer scientists can use it as a structural blueprint.

Understanding this blueprint matters immensely because it simplifies complex operations. In pure mathematics, knowing the prime factors of numbers allows you to effortlessly find the Greatest Common Divisor (GCD) and Least Common Multiple (LCM), which are essential for adding or subtracting fractions with different denominators. In the modern digital world, prime factorization is the literal lock and key of internet security. The difficulty of finding the prime factors of massively large numbers is the exact mathematical hurdle that protects your credit card information, private emails, and bank accounts from hackers. Without the unique, immutable nature of prime factorization, modern mathematics and digital infrastructure would completely collapse.

History and Origin

The conceptual roots of prime factorization stretch back to ancient Greece, specifically to the mathematician Euclid of Alexandria around 300 BC. In his seminal work, Elements (specifically Book VII), Euclid laid the groundwork for number theory by defining prime numbers and proving that there are infinitely many of them. He also formulated what is now known as Euclid's Lemma, a critical stepping stone toward understanding that prime factorization is unique. Shortly after Euclid, around 240 BC, the Greek scholar Eratosthenes developed the "Sieve of Eratosthenes," the first systematic algorithm for identifying prime numbers by iteratively marking the multiples of each prime, starting with 2. These ancient discoveries provided the raw materials for factorization, but the formalization of the concept took many more centuries.

The true mathematical codification of prime factorization arrived with the legendary German mathematician Carl Friedrich Gauss. In 1801, at the age of 24, Gauss published Disquisitiones Arithmeticae, a monumental textbook that revolutionized number theory. In this text, Gauss provided the first formal, rigorous proof of the Fundamental Theorem of Arithmetic. While earlier mathematicians essentially knew that numbers broke down into primes, Gauss proved mathematically that this breakdown is absolutely unique for every single integer. His rigorous approach transformed prime factorization from an observed phenomenon into an unshakeable law of mathematics, setting the stage for all modern number theory.

The practical application of prime factorization remained largely confined to pure mathematics until the late 20th century, when the dawn of the internet required a new method for secure communication. In 1977, researchers Ron Rivest, Adi Shamir, and Leonard Adleman at MIT introduced the RSA public-key cryptosystem. They realized that while it is computationally trivial for a computer to multiply two large prime numbers together, it is computationally devastating to reverse the process—to take the massive resulting number and figure out the two original primes. By tying encryption to the difficulty of prime factorization, RSA transformed this ancient mathematical curiosity into the foundation of global cybersecurity, ensuring that a concept explored by Euclid 2,300 years ago now protects trillions of dollars in global e-commerce.

Key Concepts and Terminology

To navigate the landscape of prime factorization, you must master the specific vocabulary used by mathematicians. The most critical term is Prime Number, which is an integer strictly greater than 1 that is divisible only by 1 and itself. Examples include 2, 3, 5, 7, 13, and 97. The number 2 is unique as the only even prime number; all other even numbers are divisible by 2 and are therefore composite. A Composite Number is any integer greater than 1 that is not prime. Composite numbers have at least three positive divisors: 1, the number itself, and at least one other positive integer.

A Factor (or Divisor) of a given number is any integer that divides into that number evenly, leaving a remainder of zero. For example, the factors of 10 are 1, 2, 5, and 10. A Prime Factor, however, is a factor that is also a prime number. In the case of 10, the prime factors are only 2 and 5. The distinction between a general factor and a prime factor is the most common stumbling block for beginners. When we perform prime factorization, we are exclusively seeking the prime factors, and we must find the exact quantity of each prime factor required to multiply together to reach the original number.

This brings us to the concept of Multiplicity, which refers to the number of times a specific prime factor appears in the factorization of a number. For example, in the prime factorization of 8 (which is 2 × 2 × 2), the prime factor 2 has a multiplicity of 3. To express multiplicity cleanly, mathematicians use Exponents (or powers). Instead of writing 2 × 2 × 2, we write $2^3$. The standard mathematical notation for the prime factorization of a number $N$ is expressed as $N = p_1^{a_1} \times p_2^{a_2} \times \dots \times p_k^{a_k}$, where $p$ represents distinct prime numbers and $a$ represents their respective exponents. Finally, two numbers are considered Coprime (or relatively prime) if they share absolutely no common prime factors. For example, 14 (factors: 2, 7) and 15 (factors: 3, 5) are coprime because their prime factorizations do not overlap, meaning their Greatest Common Divisor is exactly 1.

How It Works — Step by Step

The most reliable and universally taught method for finding the prime factorization of a number is called Trial Division. This algorithm works by systematically testing potential prime factors, starting from the smallest prime number (2) and moving upward, dividing the target number until only a prime number remains. The mathematical formula we are satisfying is $N = p_1^{a_1} \times p_2^{a_2} \times \dots \times p_k^{a_k}$. You begin by dividing your number $N$ by 2 as many times as possible until it is no longer evenly divisible by 2. Then, you move to the next prime number, 3, and divide the remaining quotient by 3 as many times as possible. You continue this process with 5, 7, 11, and so on, until the final quotient is itself a prime number or 1.

Let us perform a complete, step-by-step worked example using the number 360. Step 1: Start with the smallest prime, 2. Since 360 is even, it is divisible by 2. We calculate $360 \div 2 = 180$. We record our first prime factor: 2. Step 2: Take the new quotient, 180, and test 2 again. $180 \div 2 = 90$. We record another 2. Step 3: Take 90 and test 2 again. $90 \div 2 = 45$. We record a third 2. Step 4: Take 45 and test 2. 45 is odd, so it is not divisible by 2. We move to the next prime, 3. Step 5: Test 3. $45 \div 3 = 15$. We record a prime factor: 3. Step 6: Take 15 and test 3 again. $15 \div 3 = 5$. We record another 3. Step 7: Take the new quotient, 5. Because 5 is a prime number itself, we cannot divide it by anything other than 1 and 5. We record the final prime factor: 5. Our complete list of recorded prime factors is 2, 2, 2, 3, 3, and 5. Grouping these using exponents, the final prime factorization is $360 = 2^3 \times 3^2 \times 5^1$.

Let us look at a second, more complex example using the number 8,190. Step 1: Test 2. $8190 \div 2 = 4095$. Record 2. Step 2: Test 2 on 4095. It fails (odd number). Move to 3. To quickly check if 4095 is divisible by 3, add its digits: $4+0+9+5 = 18$. Since 18 is divisible by 3, 4095 is divisible by 3. $4095 \div 3 = 1365$. Record 3. Step 3: Test 3 on 1365. Sum of digits is $1+3+6+5 = 15$. Divisible by 3. $1365 \div 3 = 455$. Record another 3. Step 4: Test 3 on 455. Sum of digits is $4+5+5 = 14$. Fails. Move to the next prime, 5. Step 5: Test 5 on 455. Ends in 5, so it works. $455 \div 5 = 91$. Record 5. Step 6: Test 5 on 91. Fails (does not end in 0 or 5). Move to 7. Step 7: Test 7 on 91. $91 \div 7 = 13$. Record 7. Step 8: The remaining quotient is 13. Since 13 is a prime number, we stop and record 13. The prime factorization is $8190 = 2 \times 3^2 \times 5 \times 7 \times 13$. By systematically exhausting every prime number from smallest to largest, trial division guarantees an accurate and complete prime factorization every time.

Types, Variations, and Methods

While Trial Division is the most fundamental approach, mathematicians and computer scientists have developed several different methods for prime factorization, each optimized for different scenarios, human learning styles, or computational limits.

The Factor Tree Method

For beginners and visual learners, the Factor Tree is the most popular manual method. Instead of strictly starting with the smallest prime, you break the target number into any two factors that multiply to create it, drawing two branches downward. If a branch ends in a composite number, you split it again into two more branches. If a branch ends in a prime number, you circle it—that branch is finished. You continue branching until every endpoint is a circled prime number. For example, to factor 48, you might split it into 6 and 8. The 6 splits into 2 and 3 (both prime, circle them). The 8 splits into 2 and 4. The 4 splits into 2 and 2. Collecting all the circled numbers gives you 2, 3, 2, 2, and 2, which is $2^4 \times 3$. The beauty of the factor tree is that no matter which two factors you start with (you could have started with 4 and 12), the Fundamental Theorem of Arithmetic guarantees you will end up with the exact same final prime leaves.

Fermat's Factorization Method

Developed by Pierre de Fermat in the 1600s, this algebraic method is highly effective for factoring odd composite numbers that are the product of two primes relatively close to each other. Fermat's method relies on the mathematical identity that any odd number can be expressed as the difference of two perfect squares: $N = a^2 - b^2 = (a-b)(a+b)$. To use this method, you take the square root of $N$, round up to the next whole number to get your starting $a$, and check if $a^2 - N$ results in a perfect square ($b^2$). If it does not, you increment $a$ by 1 and try again. Worked Example: Factor $N = 5959$.

1. The square root of 5959 is approximately 77.19. We round up to $a = 78$.
2. Test $a = 78$: $78^2 - 5959 = 6084 - 5959 = 125$. The number 125 is not a perfect square.
3. Test $a = 79$: $79^2 - 5959 = 6241 - 5959 = 282$. Not a perfect square.
4. Test $a = 80$: $80^2 - 5959 = 6400 - 5959 = 441$. The number 441 is a perfect square (it is $21^2$).
5. Therefore, $a = 80$ and $b = 21$. The factors are $(a-b)$ and $(a+b)$, which means $(80-21) = 59$, and $(80+21) = 101$. Both 59 and 101 are prime, so the prime factorization of 5959 is $59 \times 101$.

Advanced Computational Algorithms

For massively large numbers, trial division and Fermat's method are too slow. Computer scientists use probabilistic and advanced algebraic algorithms. Pollard's rho algorithm is excellent for finding small prime factors of very large numbers by using a pseudo-random sequence and detecting cycles. For the absolute largest numbers (such as breaking RSA keys), the General Number Field Sieve (GNFS) is the fastest known algorithm. GNFS operates in sub-exponential time, using complex polynomials and matrix algebra across massive distributed computing networks to find congruences that eventually yield the prime factors. While a human will never perform GNFS on paper, it is the gold standard algorithm running under the hood of the world's most powerful prime factorization supercomputers.

Real-World Examples and Applications

The most common everyday application of prime factorization is found in basic arithmetic, specifically when finding the Least Common Multiple (LCM) and Greatest Common Divisor (GCD). Suppose a logistics manager needs to schedule maintenance for three different delivery trucks. Truck A requires maintenance every 28 days, Truck B every 42 days, and Truck C every 60 days. If they all receive maintenance today, when is the next time they will all be in the shop on the exact same day? To find this, we need the LCM of 28, 42, and 60, which we find via prime factorization.

• $28 = 2^2 \times 7$
• $42 = 2 \times 3 \times 7$
• $60 = 2^2 \times 3 \times 5$ To find the LCM, we take the highest power of every prime number present across all factorizations: $2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 = 420$. The trucks will all be in the shop together in exactly 420 days.

In engineering and mechanical design, prime factorization is vital for calculating gear ratios. If an engineer is designing a transmission with a specific gear ratio, they must ensure the gears wear evenly. If a 15-tooth gear drives a 30-tooth gear, the exact same teeth will meet every two revolutions, leading to uneven wear and quick failure. By looking at the prime factorization, engineers design gears with "coprime" numbers of teeth (e.g., a 15-tooth gear and a 31-tooth gear). Because 15 ($3 \times 5$) and 31 (prime) share no common prime factors, every single tooth on the first gear will touch every single tooth on the second gear before the cycle repeats, ensuring perfectly even mechanical wear.

The most critical modern application, however, is the RSA encryption algorithm, which secures virtually all internet traffic. RSA relies on the fact that multiplying two large primes is easy, but factoring the result is practically impossible. Here is a simplified mathematical example of RSA in action. A computer generates two prime numbers, $p = 61$ and $q = 53$. It multiplies them to create the "public key" modulus $N = 3233$. This number 3233 is broadcast to the world so anyone can encrypt a message to you. However, to decrypt the message, a hacker must know the original primes 61 and 53. With a small number like 3233, a hacker's computer can trial-divide and find 61 and 53 in milliseconds. But in the real world, $N$ is a number with 617 decimal digits (a 2048-bit key). There is no computer on Earth fast enough to find the prime factorization of a 617-digit number, keeping your encrypted data perfectly safe.

Common Mistakes and Misconceptions

The single most pervasive misconception among beginners is the belief that the number 1 is a prime number, and therefore should be included in a prime factorization. The number 1 is definitively not prime; it is categorized as a "unit." If 1 were considered a prime number, the Fundamental Theorem of Arithmetic would collapse. Factorizations would no longer be unique because you could write the factorization of 10 as $2 \times 5$, or $1 \times 2 \times 5$, or $1 \times 1 \times 1 \times 2 \times 5$ infinitely. Therefore, you must never include 1 in a prime factorization.

A second common mistake is stopping the factorization process prematurely by leaving composite numbers in the final answer. For example, when asked to find the prime factorization of 72, a student might use a factor tree, split 72 into 8 and 9, and mistakenly write $8 \times 9$ as the final answer. While 8 times 9 does equal 72, neither 8 nor 9 are prime numbers. The process is not complete until every single factor is prime. The 8 must be broken down into $2 \times 2 \times 2$, and the 9 must be broken down into $3 \times 3$, yielding the correct prime factorization of $2^3 \times 3^2$.

Another frequent pitfall involves confusing the factors of a number with the prime factorization of a number. If an exam asks for "all factors of 20," the correct answer is the list: 1, 2, 4, 5, 10, and 20. These are all the integers that divide evenly into 20. However, if the exam asks for the "prime factorization of 20," providing that list is completely wrong. The prime factorization is the specific multiplication equation $2^2 \times 5$. Understanding the distinct difference between a list of divisors and a multiplicative prime blueprint is essential for mathematical accuracy.

Finally, many people intuitively assume that large numbers must inherently have large prime factors, while small numbers have small prime factors. This is mathematically false. The massive number 1,048,576 is simply $2^{20}$; its only prime factor is the smallest prime number in existence. Conversely, the relatively small number 97 has a massive prime factor: 97 itself. The size of a composite number dictates nothing about the size of the prime numbers required to build it.

Best Practices and Expert Strategies

When performing prime factorization manually or writing code for a basic calculator, experts rely heavily on Divisibility Rules to drastically speed up the trial division process. Instead of blindly dividing numbers and checking for decimals, you should memorize these four critical rules:

1. Rule for 2: If the number ends in 0, 2, 4, 6, or 8, it is divisible by 2.
2. Rule for 3: Add all the digits of the number together. If the sum is divisible by 3, the entire number is divisible by 3. (e.g., for 5,112, the sum is $5+1+1+2 = 9$. Since 9 is divisible by 3, 5,112 is divisible by 3).
3. Rule for 5: If the number ends in 0 or 5, it is divisible by 5.
4. Rule for 11: Subtract and add the digits in an alternating pattern. If the result is 0 or divisible by 11, the number is divisible by 11. (e.g., for 2,783, calculate $2 - 7 + 8 - 3 = 0$. Therefore, 2,783 is divisible by 11).

The most powerful expert strategy, however, is the Square Root Stopping Rule. When you are using trial division to factor a number $N$, you do not need to test prime numbers all the way up to $N$. You only need to test prime numbers up to the square root of $N$ ($\sqrt{N}$). Why? Because factors always come in pairs. If a number $N$ has a prime factor larger than its square root, it must be paired with a prime factor smaller than its square root.

Let us look at a practical example: factoring the number 103. The square root of 103 is approximately 10.14. Therefore, the only prime numbers we need to test are those smaller than 10.14, which are 2, 3, 5, and 7.

• 103 is odd (fails 2).
• $1+0+3 = 4$ (fails 3).
• Does not end in 0 or 5 (fails 5).
• $103 \div 7 = 14.71$ (fails 7). Because it failed all primes up to its square root, we can stop immediately and declare with 100% mathematical certainty that 103 is a prime number. If we did not use this rule, we would waste time testing 11, 13, 17, 19, and so on, all the way up to 101. Implementing the square root limit is the difference between an amateur approach and a highly optimized, professional algorithm.

Edge Cases, Limitations, and Pitfalls

The primary limitation of prime factorization is computational complexity, often referred to as the "Factoring Bottleneck." While algorithms like trial division work perfectly in theory for any number, they fail practically when numbers become exceptionally large. The time required to factor a number scales exponentially with the number of digits. If you have a 100-digit number that is a "semiprime" (the product of exactly two large prime numbers), a standard desktop computer using trial division would take longer than the current age of the universe to find the two factors. This asymmetrical difficulty—easy to multiply, nearly impossible to factor—is exactly why it is used in cryptography, but it represents a hard limitation for mathematicians trying to analyze large datasets.

A notable edge case occurs when dealing with negative numbers. Strictly speaking, prime numbers are defined as positive integers greater than 1, meaning the concept of prime factorization applies exclusively to positive integers. However, in advanced algebra, you may be required to factor a negative number like -42. The industry standard workaround is to extract -1 as a distinct unit factor, and then perform standard prime factorization on the absolute value. The factorization of -42 would be written as $-1 \times 2 \times 3 \times 7$. It is vital to note that -1 is not a prime number; it is a unit multiplier used to satisfy the negative sign.

The most looming pitfall for the future of prime factorization is the advent of quantum computing. In 1994, mathematician Peter Shor developed "Shor's Algorithm," a quantum computer algorithm capable of finding the prime factors of an integer in polynomial time rather than exponential time. In simple terms, while a classical supercomputer might take billions of years to factor a 2048-bit RSA key, a sufficiently powerful quantum computer running Shor's Algorithm could theoretically factor it in hours or minutes. Currently, this is an edge case because quantum computers with enough stable "qubits" do not yet exist. However, the mathematical proof of Shor's Algorithm means that the computational limitation of prime factorization is a hardware problem, not an absolute mathematical barrier, forcing the cybersecurity industry to begin developing "post-quantum" cryptographic algorithms.

Industry Standards and Benchmarks

In the realms of cryptography and computer science, prime factorization benchmarks are strictly monitored by organizations like the National Institute of Standards and Technology (NIST) and the RSA Factoring Challenge. The standard metric for measuring the difficulty of a factorization problem is the "bit length" of the number being factored. A bit is a binary digit (0 or 1), and a 1024-bit number translates to a decimal number with approximately 309 digits.

Historically, the RSA Factoring Challenge offered cash prizes to anyone who could factor specific massive semiprimes. In 2009, a team of researchers successfully factored RSA-768 (a 232-digit number) using the General Number Field Sieve, taking the equivalent of 2,000 years of computing on a single core. In February 2020, researchers set a new world record by factoring RSA-250, a 250-decimal-digit number (829 bits). This monumental effort required approximately 2,700 CPU years of computing power distributed across global supercomputer clusters.

Because of these benchmarks, NIST has established strict industry standards for digital security. As of the 2020s, a 1024-bit key is considered mathematically "broken" and unsafe for commercial use. The absolute minimum industry standard for RSA encryption is a 2048-bit key (a number with 617 decimal digits). For highly sensitive military or financial data, a 4096-bit key is standard. These benchmarks rely entirely on the premise that classical prime factorization algorithms scale at sub-exponential time complexity, specifically $O(\exp((\frac{64}{9} b)^{\frac{1}{3}} (\log b)^{\frac{2}{3}}))$ for the General Number Field Sieve, where $b$ is the number of bits. Understanding these benchmarks allows developers to choose key sizes that will remain unfactorable by classical computers for decades to come.

Comparisons with Alternatives

When discussing prime factorization, it is crucial to distinguish it from related mathematical processes, specifically Standard Factoring and Primality Testing.

Standard factoring involves finding every integer that divides evenly into a number, not just the prime ones. If a teacher asks for the standard factors of 24, you must find 1, 2, 3, 4, 6, 8, 12, and 24. This is usually done by finding factor pairs ($1 \times 24$, $2 \times 12$, $3 \times 8$, $4 \times 6$). Prime factorization, on the other hand, breaks the number down into its deepest elemental state: $2^3 \times 3$. You would choose standard factoring when trying to divide a real-world quantity into equal groups (e.g., splitting 24 students into 4 groups of 6). You would choose prime factorization when performing advanced operations like finding a common denominator for complex fractions or simplifying square roots. Notably, once you have the prime factorization, you can easily calculate the total number of standard factors by adding 1 to each exponent and multiplying them together. For 24 ($2^3 \times 3^1$), the exponents are 3 and 1. $(3+1) \times (1+1) = 4 \times 2 = 8$ total factors.

Primality Testing is another alternative process that answers a simpler question: "Is this number prime, yes or no?" Algorithms like the Miller-Rabin primality test or the AKS primality test do not tell you what the factors of a number are; they only tell you if the number is composite or prime. Primality testing is vastly faster than prime factorization. A computer can use the Miller-Rabin test to prove that a 1000-digit number is prime in a fraction of a second. However, if the test reveals the number is composite, it gives you no clues as to what the prime factors actually are. You would choose primality testing when generating keys for RSA (where you simply need to find two large, verified prime numbers), whereas you would use prime factorization if you were a hacker attempting to break those keys.

Frequently Asked Questions

Is the number 1 a prime number, and should it be in my prime factorization? No, the number 1 is definitively not a prime number; it is classified mathematically as a "unit." A prime number must have exactly two distinct positive divisors: 1 and itself. The number 1 only has one divisor. More importantly, if 1 were included in prime factorizations, the Fundamental Theorem of Arithmetic would fail, because a factorization would no longer be unique (e.g., 14 could be $2 \times 7$, or $1 \times 2 \times 7$, or $1 \times 1 \times 2 \times 7$). Therefore, 1 is never included in a prime factorization.

Can prime factorization be applied to negative numbers or decimals? Strictly speaking, prime factorization applies only to positive integers greater than 1. Decimals and fractions cannot be prime factored because the concept relies on whole-number divisibility. For negative integers, the mathematical convention is to extract -1 as a unit multiplier, and then perform standard prime factorization on the positive absolute value. For example, the factorization of -30 is expressed algebraically as $-1 \times 2 \times 3 \times 5$.

What is the fastest way to find the prime factorization of a large number by hand? The fastest manual method combines the Square Root Stopping Rule with Divisibility Rules. First, calculate the approximate square root of your target number to establish the maximum prime you need to test. Then, use divisibility rules for 2, 3, 5, and 11 to quickly eliminate small factors without doing long division. If you extract a factor, immediately calculate the square root of the new smaller quotient to further lower your testing ceiling. This prevents you from wasting time testing large primes unnecessarily.

How do I know if I have finished a factor tree correctly? A factor tree is complete when every single endpoint (or "leaf") at the bottom of the branches is a prime number. To verify your work, you must do two things: First, check each endpoint to ensure it is not a composite number (for example, make sure you didn't leave a 9 instead of breaking it into 3 and 3). Second, multiply all the endpoints together. If the product equals your original starting number, and all numbers are prime, your prime factorization is 100% correct.

Why do we use exponents in prime factorization? Exponents are used for clarity, brevity, and advanced mathematical calculations. When factoring a number like 1,024, writing out $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ is cumbersome and prone to copying errors. Writing $2^{10}$ is concise. Furthermore, having the factorization in exponential form allows mathematicians to use exponent rules to instantly calculate the total number of divisors a number has, or to quickly find Greatest Common Divisors by comparing the exponent values of shared primes.

What happens if a number is already prime? What is its prime factorization? If a number is already prime, its prime factorization is simply the number itself. You do not write "1 times the number," because 1 is not prime. For example, the prime factorization of 17 is just 17. In exponential notation, you could write it as $17^1$. This satisfies the rule that the number is represented entirely by prime numbers.

How is prime factorization used to simplify square roots? Prime factorization is the most reliable method for simplifying radical expressions. To simplify $\sqrt{72}$, you first find the prime factorization: $2 \times 2 \times 2 \times 3 \times 3$. Because a square root asks for pairs of identical numbers, you group the factors into pairs. We have one pair of 2s and one pair of 3s, with a single 2 left over. For every pair, you pull one number outside the radical. Pull out a 2 and a 3 ($2 \times 3 = 6$), and leave the unpaired 2 inside. The simplified exact answer is $6\sqrt{2}$.