Math101 · Independent Ontario learning library
Math101learn.math101.caPrime Factorization
Prime factorization expresses an integer greater than $1$ as a product of prime numbers. Apart from the order of factors, this product is unique—the Fundamental Theorem of Arithmetic.
Unique prime building blocks let many number problems become comparisons of exponents. They make fraction reduction and common-factor calculations transparent rather than trial-based.
Intuition and core definition
Prime factorization expresses an integer greater than $1$ as a product of prime numbers. Apart from the order of factors, this product is unique—the Fundamental Theorem of Arithmetic. For example, $180=2^2\cdot3^2\cdot5$. Prime factorization reveals the building blocks used in divisibility, GCF, LCM, radicals, and fractions.
Notation, language, and conditions
Exponents compress repeated prime factors: $2^3$ means $2\cdot2\cdot2$. A factor tree may branch in different ways, but complete trees end with the same prime leaves. The number $1$ has an empty prime factorization and is not prime; negative integers can be written as $-1$ times the factorization of their magnitude.
Why this idea matters
Prime factorization expresses a whole number through indivisible multiplicative building blocks and is unique apart from factor order.
A dependable method
- Divide by the smallest prime that works, or split the number into any convenient factor pair.
- Continue factoring every composite result.
- Stop only when all remaining factors are prime.
- Collect identical primes and write them with exponents in increasing order.
- Multiply the factors to reconstruct the original number.
Worked example
Representations and interpretation
A factor tree records multiplicative decomposition, while an exponent vector records how many copies of each prime occur. Different tree shapes represent different routes to the same unique endpoint.
Reasoning about variations
A perfect square has only even exponents in its prime factorization. Thus $2^4\cdot3^2$ is a square, while $2^3\cdot3^2$ is not. This observation later makes radical simplification systematic.
Common mistakes
How to check your work
- Multiply the prime powers and recover the original integer.
- Confirm every base in the final product is prime.
- Use divisibility: the exponents must support every known divisor of the original number.
Practice
- Find the prime factorization of $360$.
- Is $2^4\cdot5^2$ a perfect square?
- What is wrong with $84=4\cdot3\cdot7$ as a final prime factorization?
Answers and brief solutions
Show answers
- $2^3\cdot3^2\cdot5$ $360=36\cdot10=(2^2\cdot3^2)(2\cdot5)$.
- Yes Both prime exponents are even.
- $4$ is composite $4$ must be replaced by $2^2$.
Synthesis and transfer
Reducing a large fraction can be organized by prime-exponent columns: shared minimum exponents cancel, while the remaining exponents reconstruct the simplified numerator and denominator.
Write each number as a vector of prime exponents. For $360=2^3\cdot3^2\cdot5$ and $84=2^2\cdot3\cdot7$, their GCF uses componentwise minima, $2^2\cdot3=12$, while their LCM uses maxima, $2^3\cdot3^2\cdot5\cdot7=2520$. This one representation therefore supports fraction reduction, shared grouping, and cycle alignment. Multiplying the recorded prime powers must recover the original number, which checks transcription and exponent counts. A factor tree may branch in many ways, but continuing every composite leaf eventually yields the same prime multiset; uniqueness is what makes exponent bookkeeping reliable.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find the prime factorization of $360$.
- $360=36\cdot10=(2^2\cdot3^2)(2\cdot5)$.
End of lesson
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