What are the factors of 12?

12 has 6 divisors in total: 1, 2, 3, 4, 6, 12. Its prime factorisation is 2^2 × 3, and every one of those 6 divisors is built from those primes. The divisor count follows directly from the exponents: (2 + 1) × (1 + 1) = 6. They sum to 28.

What is the nearest prime to 12?

The prime immediately below 12 is 11, and the one immediately above is 13. Since 12 is composite it sits inside the run between them, 1 number wide.

What is Euler's totient of 12?

φ(12) = 4. The totient counts how many numbers below 12 share no common factor with it. Computing it from the factorisation 2^2 × 3 gives 12 × (1 − 1/2) × (1 − 1/3) = 4.

What does the totient function count?

How many numbers below n share no common factor with it. For a prime p the answer is p − 1, because every smaller number is coprime to it. For composites it is smaller, computed as n multiplied by (1 − 1/p) across each distinct prime factor. It is the function at the heart of RSA encryption, which is why a piece of pure number theory from 1763 now protects internet traffic.

Why do prime factors matter outside mathematics?

Because multiplying two large primes is easy and undoing it is not, and nearly all public-key cryptography rests on that gap. RSA multiplies two primes of several hundred digits and publishes the product; recovering the factors is believed infeasible at that size. Small numbers factorise instantly, which is exactly why real keys are enormous.

What is a Mersenne prime?

A prime of the form 2^p − 1 where p is itself prime. 3, 7, 31 and 127 are the first four. The exponent being prime is necessary but nowhere near sufficient — 2¹¹ − 1 is 2,047, which is 23 × 89. They matter because there is a fast test that works only for numbers of this shape, which is why every record-breaking largest known prime for decades has been a Mersenne prime.

How does the number of divisors work?

It comes from the exponents in the factorisation, not from the size of the number. Add one to each exponent and multiply: 12 is 2² × 3, so it has (2+1) × (1+1) = 6 divisors. This is why a small number packed with low prime powers can have more divisors than a much larger one — 48 has ten, while 47 has two.

Is there a formula that produces only primes?

Nothing useful. Euler noticed that n² + n + 41 gives a prime for every n from 0 to 39, which is remarkable, but it fails at n = 40 where the result is 41². No polynomial with integer coefficients can produce primes for every input — it is a theorem, not merely an absence of cleverness. Formulas that do exist are far too slow to compute.

Is 12 a Prime Number?

12 is the smallest number with six divisors, and the reason clocks, months, inches and dozens are counted the way they are. Divisibility by 2, 3, 4 and 6 makes it far more useful for splitting things than the larger and primer 11 or 13 on either side. It is not prime. 12 = 2 × 2 × 3, so it has 6 div...

YOUR NUMBER

NUMBER

WORTH TRYING

12 IS

NOT PRIME

SMALLEST FACTOR

12 = 2^2 × 3

Prime factorisation

2^2 × 3

Number of divisors

Sum of divisors

Totient φ(n)

Previous prime

Next prime

FACTOR BREAKDOWN

PRIME

POWER

CONTRIBUTES

STEPS

Trial-divide 12 by every prime up to √12 ≈ 3 — beyond that a factor would have a partner below it

The smallest prime divisor is 2, so 12 is composite

Full factorisation: 2^2 × 3

12 is NOT prime

SIEVE OF ERATOSTHENES

Two thousand years old and still the fastest way. Take the first uncrossed number, 2, and strike out every multiple of it. Move to the next survivor, 3, and repeat. Each pass can start at the square of its prime, because anything smaller was already struck by a smaller factor — which is why the passes get shorter and stop entirely once the prime exceeds √100. Whatever survives is prime. Your number 12 is outlined in gold.

HOW MANY PRIMES BELOW x — AND WHAT THE THEORY PREDICTS

4 vs 4

100

25 vs 22

1,000

168 vs 145

10,000

1,229 vs 1,086

100,000

9,592 vs 8,686

Actual count against x ÷ ln x. At x = 10 the estimate overshoots slightly; from 100 upward it runs low, and from 1,000 onward the gap narrows steadily — a ratio of 1.16 at a thousand down to 1.10 at a hundred thousand. That it tends to 1 at all is the Prime Number Theorem, and how slowly it does so is why the proof took a century to find.

Proved, not guessed. Primality here uses a deterministic Miller–Rabin test with a fixed witness set that is proven correct for every number this page accepts, so a result of PRIME is a proof rather than a probability. Ranges are sieved up to 200,000; single numbers are tested up to 4,503,599,627,370,495.

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HOW TO USE

1
The answer is already on screen: 12 is not prime, because 12 = 2 × 2 × 3. Nothing needs typing — this page opens the calculator with 12 loaded.
2
Read the STEPS panel to see the reasoning: the smallest prime divisor of 12 is 2, which is enough to settle it, and the full factorisation 2^2 × 3 follows.
3
Check the six tiles for the derived quantities — divisor count, divisor sum, Euler's totient and the primes either side. All of them come from the factorisation rather than from separate searches.
4
Type a different number to carry on, or switch to RANGE to sieve a whole interval at once and see how the primes thin out as the numbers grow.

THE FORMULAS

Primeexactly two divisors: 1 and n

Trial division limittest primes up to √n only

Fundamental theoremn = p₁^a¹ × p₂^a² × … uniquely

Number of divisorsd(n) = Π (aᵢ + 1)

Sum of divisorsσ(n) = Π (pᵢ^(aᵢ+1) − 1)/(pᵢ − 1)

Euler totientφ(n) = n · Π (1 − 1/pᵢ)

Totient of a primeφ(p) = p − 1

Sieve pass startbegin each pass at p²

Prime Number Theoremπ(x) ≈ x ÷ ln x

Twin primesp and p + 2 both prime

Mersenne prime2^p − 1, with p itself prime

Perfect numberσ(n) − n = n

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Last updated: July 28, 2026 · Formula verified · Deterministic Miller–Rabin, so a result of PRIME is proven rather than probable · Eagle-eyed accuracy for every calculation.

Is 12 a Prime Number?

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HOW TO USE

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