What is the prime factorization of 360?

What Is Prime Factorization?

Prime factorization expresses a composite number as a product of prime numbers that multiply back to the original. Every integer greater than 1 can be written this way, and the result is always the same no matter which path you take — the fundamental theorem of arithmetic guarantees exactly one prime factorization for every number. For 360 the answer is 2³ × 3² × 5, meaning three 2s, two 3s and one 5 multiply back to 360.

Two layouts show the same work. The division ladder keeps dividing at the top and writing each prime down the side, while a factor tree splits the number into two branches and keeps branching until every leaf is prime. Either way the primes you collect are identical — only the shape of the working changes. Prime factorization is also how a number's full factor list is built: choose how many 2s, how many 3s and how many 5s to include, and every combination is a factor.

How to Use This Prime Factorization Calculator (2026)

1. Enter your number. Type any whole number from 1 up to 1 billion (or click a preset chip like 360 or 2026).
2. Click Calculate. You get the prime factorization in exponent form, the expanded product of primes, and a breakdown of distinct primes, total prime factors and the largest prime factor.
3. Read the steps. Every result includes the division ladder — each division from the original number down to 1 — so you can repeat the same working by hand on homework.
4. Check the answers. The result also tells you whether the number itself is prime, so you can verify prime or composite classifications instantly.

How to Find Prime Factorization: The Division Ladder

The fastest method by hand is repeated division: always divide by the smallest prime that fits the current quotient, write that prime down, and continue until the quotient reaches 1. Worked example for 360:

Step	Division	Quotient
1	360 ÷ 2	180
2	180 ÷ 2	90
3	90 ÷ 2	45
4	45 ÷ 3	15
5	15 ÷ 3	5
6	5 ÷ 5	1

The divisors down the left are three 2s, two 3s and one 5, so 360 = 2³ × 3² × 5. Exponent form is shorthand for the expanded product 2 × 2 × 2 × 3 × 3 × 5, and it is the form you need for the divisor count: (3+1)(2+1)(1+1) = 24 factors of 360 in all — list them with our factor calculator.

Prime Factorization of Common Numbers — Reference Table

Quick lookup for the numbers that show up most in homework. The distinct-primes column counts how many different primes appear; the total column counts every prime with multiplicity. Both come straight out of the exponent form.

Number	Prime Factorization	Distinct Primes	Total Prime Factors
12	2² × 3	2	3
100	2² × 5²	2	4
120	2³ × 3 × 5	3	5
360	2³ × 3² × 5	3	6
720	2&sup4; × 3² × 5	4	7
1000	2³ × 5³	2	6
2310	2 × 3 × 5 × 7 × 11	5	5
97	97 (prime)	1	1
1	none — 1 has no prime factors	0	0

Notice 2310 = 2 × 3 × 5 × 7 × 11 uses the first five primes exactly once — it is the product of the first five primes, and no smaller number has five distinct prime factors. Multiply those primes back together in any combination and you have built every one of its 32 factors.

Factor Trees: The Other Way to See It

A factor tree makes the same splits visible as branches. Start with the target number at the top and split it into any two factors, then keep splitting composite branches until every leaf is prime. One tree for 360:

You could split 360 as 36 × 10, or 4 × 90, or 2 × 180 first — every valid tree ends with the same leaves, because the primes of a number never change. That invariance is exactly what the fundamental theorem of arithmetic states.

Why Prime Factorization Is Unique

The fundamental theorem of arithmetic says every integer greater than 1 is either prime itself or can be written as a product of primes in exactly one way, ignoring the order of the factors. 2 × 3 × 2 × 3 × 2 and 2³ × 3² are the same factorization written differently; no other set of primes multiplies to 360. This is why 1 is excluded from the definition of prime — if 1 counted as prime, 12 could be written 2² × 3, or 2² × 3 × 1, or 2² × 3 × 1 × 1, and uniqueness would collapse.

Prime numbers are the edge case: the prime factorization of 97 is simply 97, and the number 1 has none at all — it is neither prime nor composite. Every other number has at least two prime factors to collect.

How to Find Prime Factorization on a Calculator

Handheld calculators have no dedicated “prime factorize” button, so the workaround is repeated division: divide by the smallest prime that fits, write it down, and repeat with the quotient. On any scientific calculator:

1. Divide by 2 while the quotient is even: 360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45.
2. Move to 3, then 5, then the next primes — 45 ÷ 3 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1.
3. Collect the divisors: 360 = 2³ × 3² × 5. Check by multiplying the primes back together.

Graphing calculators such as the TI-84 loop the same division in a program, while CAS models (TI-89, TI-Nspire CAS, Casio ClassPad) factor directly with a built-in command. For factoring polynomials rather than numbers — by hand, on a TI-84 or by reading x-intercepts — see our trinomial factor calculator.

What Prime Factorization Is Used For

• Cryptography: RSA encryption depends on the difficulty of factoring a 2,048-bit number into two large primes — the same operation this calculator performs, scaled past any brute-force attempt.
• GCF and LCM: comparing prime factorizations gives the greatest common factor (shared primes to their lowest powers) and the least common multiple (highest powers) in one pass — try it on our GCF and LCM calculator.
• Counting factors: raise each exponent by 1 and multiply: 360 = 2³ × 3² × 5 has (3+1)(2+1)(1+1) = 24 factors without listing them — see the full list with our factor calculator.
• Reducing fractions: every fraction in lowest terms has been divided through by the shared primes in its numerator and denominator.

FAQ

Sources: Common Core State Standards for Mathematics (6.NS.B.4), NGA Center and CCSSO; National Council of Teachers of Mathematics, Principles to Actions; D. M. Rosen, Elementary Number Theory; National Institute of Standards and Technology, RSA cryptosystem specifications. This tool is for educational purposes only.