Math101 · Independent Ontario learning library
Math101Prime 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.
