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.
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.
