Prime Factorization Explained
ByMuhammad Ali•Founder of KruskalCode
12:11
6 min read
Prime factorisation (UK spelling) or prime factorization (US) is a core skill from middle school through GCSE and early algebra: rewrite an integer as a product of primes raised to powers. This article outlines the idea without heavy jargon and points to our calculator page for fast, transparent checks once you understand the rule.
Use the free calculator
Skip the longhand math — open the matching ProMathTools calculator and get an instant result.
Use the calculator →Explanation
A prime number has exactly two positive divisors—1 and itself. The Fundamental Theorem of Arithmetic says every integer n > 1 can be written uniquely as a product of primes if you ignore order. Powers save space: repeated division by the same prime becomes an exponent, so structured answers match textbook mark schemes in both metric-first UK courses and typical US curricula.
Formula
For primes p1,…,pk and positive integers a1,…,ak: n = p1^a1 × p2^a2 × … × pk^ak. Uniqueness means there is only one multiset of primes and exponents that works for each n.
Example
Take 360. Divide out 2 until it no longer fits: 360 → 180 → 90 → 45, so three factors of 2. Then 45 ÷ 3 → 15 → 5 gives two factors of 3. Finally 5 is prime. Hence 360 = 2³ × 3² × 5. The exponent line is what many examiners expect after a factor tree.
How to use the related calculator
Open the Prime Factorization calculator, type one whole number ≥ 2 in the Positive integer box, then read the printed lines: the product-of-primes string with caret exponents, how many distinct primes appear, and the total prime factors counted with multiplicity (adds duplicate primes like the three copies of 2 in 360). If you leave decimals, negatives, or values under 2, you will see a plain-English correction instead of a result.
Try the related calculator
Open toolFAQ
Is factorisation spelling wrong if I write factorization?
Both spellings are normal—factorisation is common in the UK; factorization is common in the US. The mathematics is identical.
Can primes have more than one factorization?
Only the order may change—e.g. 2×3×5 or 5×2×3. The set of primes and their exponents is unique for n > 1.
How does this help with LCM and GCD?
Write two numbers in prime-power form, then take the minimum exponent for each prime for GCD and the maximum for LCM. Our GCD and LCM & HCF pages build on that idea.
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About the author
Muhammad Ali. Muhammad Ali is a full-stack developer and founder of KruskalCode. He builds SaaS platforms and automation systems with React and Laravel, and helps teams ship fast, scalable tools.
Need a custom calculator, dashboard, or automation workflow? Reach out to KruskalCode.
