What are the factors of 13?

13 has 2 divisors in total: 1, 13. Because 13 is prime, that list can only ever be 1 and the number itself — a prime has no others by definition. They sum to 14.

What is the nearest prime to 13?

The prime immediately below 13 is 11, and the one immediately above is 17. That puts 13 in a gap of 2 below and 4 above.

What is Euler's totient of 13?

φ(13) = 12. For any prime the totient is simply the number minus one, because every smaller positive integer is coprime to it — so all 12 numbers from 1 to 12 share no factor with 13.

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.

How large can a gap between primes be?

As large as you like. Take any number n and consider the run from n! + 2 up to n! + n: every one of them is composite, because n! + k is divisible by k. That constructs a gap of any length you choose. Gaps do appear early too — there is nothing between 23 and 29, and nothing between 89 and 97.

Is 13 a Prime Number?

13 is a prime with a reputation it did nothing to earn, and the upper half of the twin pair with 11. It is also the sixth prime, and the first prime that leaves a gap of four before the next one — the first sign that primes do not keep a steady rhythm. And it is prime: exactly two divisors, 1 and 13...

YOUR NUMBER

NUMBER

WORTH TRYING

13 IS

PRIME

PRIME GAP TO NEXT

Exactly two divisors, 1 and itself. Previous prime 11, next 17. Twin prime with 11.

Prime factorisation

Number of divisors

Sum of divisors

Totient φ(n)

Previous prime

Next prime

FACTOR BREAKDOWN

PRIME

POWER

CONTRIBUTES

STEPS

Trial-divide 13 by every prime up to √13 ≈ 3 — beyond that a factor would have a partner below it

No divisor found, so 13 has exactly two divisors: 1 and itself

Every prime above 3 sits next to a multiple of 6 — here 13 is 6 × 2 + 1

13 is 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 13 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: 13 is prime. Nothing needs typing — this page opens the calculator with 13 loaded.
2
Read the STEPS panel to see why. It trial-divides by every prime up to √13 ≈ 3.61, finds nothing, and stops — checking further could only rediscover factor pairs already seen.
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 29, 2026 · Formula verified · Deterministic Miller–Rabin, so a result of PRIME is proven rather than probable · Eagle-eyed accuracy for every calculation.

Is 13 a Prime Number?

YOUR NUMBER

HOW TO USE

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