What is prime factorisation?

Writing a number as a product of primes. Every whole number above 1 can be written this way, and — more importantly — in only one way, up to the order the factors are written in. This is the fundamental theorem of arithmetic. The uniqueness is what makes the factorisation worth having: it is a fingerprint for the number, and almost every question about divisibility can be answered from it.

Why is 1 not a prime number?

Because it would destroy the uniqueness of the factorisation. If 1 counted as prime, then 6 would be 2×3, and also 1×2×3, and also 1×1×2×3, and so on forever — infinitely many factorisations of the same number. Every theorem that relies on there being exactly one would need rewriting with an awkward exception. Excluding 1 is not an arbitrary convention; it is what keeps the fundamental theorem true.

How far do I have to test before a number is prime?

Only to the square root. If n has a factor pair a and b with a no bigger than b, then a×a is at most a×b, which is n — so a is at most √n. Every composite therefore has a factor at or below its square root, and finding none there proves the number prime. For 9409 the smallest factor is 97, which is exactly the root: the bound is tight and cannot be improved.

How do I count the divisors without listing them?

Add one to each exponent and multiply. For 360 = 2³ × 3² × 5 that is 4 × 3 × 2 = 24. The reasoning is that a divisor is built by choosing how many of each prime to include: none to three twos, none to two threes, none to one five — and independent choices multiply. It scales to numbers where listing is impractical: 963761198400 has 6720 divisors, and the multiplication takes no longer than the small case.

How do I find the gcd and lcm from factorisations?

Take the smaller exponent of each prime for the gcd and the larger for the lcm. A prime missing from one number counts as exponent zero, so it contributes nothing to the gcd and its full power to the lcm. The two are linked: gcd × lcm always equals the product of the original numbers, which makes an excellent check — if it fails, an exponent has been misread somewhere.

What is Euler’s totient?

The count of numbers below n that share no factor with it. From the factorisation it is the product of pᵃ⁻¹(p − 1) over the primes. For a prime p it is simply p − 1, since everything below a prime is coprime to it. The function matters well beyond number theory: RSA encryption chooses its keys using the totient of the product of two large primes, which is easy to compute if you know the primes and hard otherwise.

Why is factoring large numbers difficult?

Because the obvious method scales with the square root of the number, not its length. A sixteen-digit number needs about 10⁸ trial divisions; adding a few digits multiplies that. Better algorithms exist — Pollard’s rho, the quadratic sieve, the number field sieve — but none is fast in the sense that multiplication is fast. That asymmetry is deliberate in cryptography: multiplying two 300-digit primes takes no time, and recovering them from the product is beyond any current machine.

How can I tell if a number is a perfect square?

Every exponent in its factorisation is even. A square is some number multiplied by itself, which doubles every exponent — so 144 = 2⁴ × 3² is a square, and its root is found by halving each exponent to get 2² × 3 = 12. The same rule extends: a perfect cube has every exponent divisible by three, and an nth power has every exponent divisible by n.

What is a perfect number?

One whose divisors, excluding itself, add up to it — equivalently, one where the sum of all divisors is exactly twice the number. The first four are 6, 28, 496 and 8128, and each has the form 2^(p−1)(2^p − 1) with the second factor prime. Euclid found that form and Euler proved every even perfect number takes it. Whether an odd perfect number exists is still unknown after more than two thousand years.

What is the largest number this page can factor?

9,007,199,254,740,991, which is 2⁵³ − 1 — the last integer a browser can hold exactly. Past that, consecutive whole numbers stop being individually representable, so a factorisation would belong to a number slightly different from the one entered. Within the cap the method is Miller–Rabin for primality and Pollard’s rho for splitting, which handles even the hardest cases in a few milliseconds rather than the minutes trial division would need.

Prime Factorization Calculator

Every whole number above 1 breaks into primes in exactly one way. That uniqueness is the whole content of the fundamental theorem of arithmetic, and it is less obvious than it sounds — start a factor tree from a different branch and every level looks different, yet the leaves always come out the sam...

The factorisation is unique, and that is why 1 is not prime. Every integer above 1 breaks into primes in exactly one way. Admit 1 as a prime and 6 becomes 2×3, then 1×2×3, then 1×1×2×3, without end — the uniqueness the whole theorem rests on would be gone.

THE NUMBER

second number, for gcd and lcm

360 =

2^3 × 3^2 × 5

3 distinct primes, 6 factors counted with multiplicity. Multiplying them back gives 360 — the same number, which is the check worth doing.

DIVISORS

4 × 3 × 2

SUM OF DIVISORS

1,170

EULER TOTIENT φ(n)

coprime to n below n

√n

the search never needs to go past this

The divisor count comes from the exponents alone. Each prime can appear anywhere from none to all of its available copies, so the choices multiply: 3 + 1, then 2 + 1, then 1 + 1 — giving 4 × 3 × 2 = 24. Nothing needs listing, which matters once the count runs into thousands.

BREAKING IT DOWN

Any order of splitting reaches the same primes. Start with a different factor and the tree looks different at every level, yet the leaves are identical. That is the uniqueness the fundamental theorem asserts, and it is far less obvious than it sounds — it took until Euclid to prove and until Gauss to state properly.

Miller–Rabin and Pollard’s rho · exact to 2⁵³ − 1

Created with❤️byeaglecalculator.com

HOW TO USE

1
Type any whole number from 2 up to 9,007,199,254,740,991. Commas and spaces are ignored; decimals are rejected, because primes are built from divisibility and a fraction divides nothing evenly.
2
Read the exponents as well as the primes. Multiplying each one plus one gives the divisor count directly, which is the quickest thing the factorisation tells you.
3
Check the reconstruction. Multiplying the factors back must give the number you typed — the page shows it, and that single line catches any error the rest of the working could hide.
4
Add a second number to get the gcd and lcm from the exponents: the smaller of each pair for one, the larger for the other. Their product always equals the product of the two numbers.

REFERENCE FORMULAS

Rule	Formula	What it is for
Fundamental theorem	n = p₁^a₁ · p₂^a₂ · … , uniquely	Every integer above 1 factors one way only, up to the order of the factors.
Why 1 is excluded	6 = 2·3 = 1·2·3 = 1·1·2·3 …	Counting 1 as prime would give infinitely many factorisations and kill uniqueness.
Trial division bound	test only up to √n	If n = ab with a ≤ b then a ≤ √n, so a composite always has a factor below the root.
Number of divisors	d(n) = ∏(aᵢ + 1)	Each prime can appear 0 to aᵢ times, so the choices multiply. No listing needed.
Sum of divisors	σ(n) = ∏ (pᵢ^(aᵢ+1) − 1)/(pᵢ − 1)	A geometric series per prime, multiplied together.
Euler totient	φ(n) = ∏ pᵢ^(aᵢ−1)(pᵢ − 1)	How many numbers below n share no factor with it. Central to RSA.
GCD from exponents	take the smaller exponent of each prime	Any prime missing from either number contributes nothing.
LCM from exponents	take the larger exponent of each prime	And gcd × lcm = a × b always, which is a quick check.
Perfect square	every exponent is even	Because a square is its root multiplied by itself, doubling every exponent.
Perfect number	σ(n) = B2	6, 28, 496, 8128. Whether an odd one exists is still unknown.
Coprime	gcd(a, b) = 1	No prime in common. Their factorisations share nothing at all.
Prime counting	π(x) ≈ x / ln x	Roughly one number in ln x is prime near x, so primes thin out slowly.
Cube of a prime	p³ has exactly 4 divisors	1, p, p², p³. From (3 + 1) — the exponent rule at its simplest.
Why factoring is hard	√n divisions for a naive search	A 16-digit number needs ~10⁸ steps. Public-key cryptography rests on this.

FREQUENTLY ASKED QUESTIONS

RELATED CALCULATORS

6 to the Power of 9

Calculate instantly →

5 to the Power of 1

Calculate instantly →

2 to the Power of 10

Calculate instantly →

5 to the Power of 4

Calculate instantly →

MORE ALGEBRA CALCULATORS

24 Times Table

Calculate instantly →

Quadratic Equation Solver

Calculate instantly →

Logarithm Calculator

Calculate instantly →

Fraction Calculator

Calculate instantly →

Fraction Simplifier

Calculate instantly →

Linear Equation Solver

Calculate instantly →

25 Times Table

Calculate instantly →

26 Times Table

Calculate instantly →

27 Times Table

Calculate instantly →

Simultaneous Equations Solver (2×2)

Calculate instantly →

28 Times Table

Calculate instantly →

Exponent Calculator

Calculate instantly →

29 Times Table

Calculate instantly →

30 Times Table

Calculate instantly →

Times Tables Mega Calculator

Calculate instantly →

Square Root of 10

Calculate instantly →

Square Root of 1

Calculate instantly →

Square Root Mega

Calculate instantly →

Square Root of 2

Calculate instantly →

23 Times Table

Calculate instantly →

Was this calculator helpful?

Last updated: August 4, 2026 · Miller–Rabin primality with Pollard’s rho, exact to 2⁵³ − 1 · The factorisation is unique, which is why 1 is not counted as prime.