What are the factors of 21?

21 has 4 divisors in total: 1, 3, 7, 21. Its prime factorisation is 3 × 7, and every one of those 4 divisors is built from those primes. The divisor count follows directly from the exponents: (1 + 1) × (1 + 1) = 4. They sum to 32.

What is the nearest prime to 21?

The prime immediately below 21 is 19, and the one immediately above is 23. Since 21 is composite it sits inside the run between them, 3 numbers wide.

What is Euler's totient of 21?

φ(21) = 12. The totient counts how many numbers below 21 share no common factor with it. Computing it from the factorisation 3 × 7 gives 21 × (1 − 1/3) × (1 − 1/7) = 12.

Because counting it would break unique factorisation. If 1 were prime then 12 could be written as 2² × 3, or 1 × 2² × 3, or 1⁹ × 2² × 3, and the promise that every number factorises in exactly one way would be false. Excluding 1 keeps that promise, and matches the two-divisor definition.

Which composites look most like primes?

The odd ones whose smallest factor is 7 or above, because every quick test passes them. 49 is 7², 77 is 7 × 11 and 91 is 7 × 13, and all three read as prime to most people. The lesson is the square-root rule: stopping early because the easy tests passed is precisely how these slip through.

Do primes thin out as numbers get larger?

Yes, steadily. There are 25 primes below 100 and 168 below 1,000, so the density falls from a quarter to under a sixth. The Prime Number Theorem makes it precise: the count below x approaches x divided by the natural logarithm of x. They never stop entirely, but they do get further apart on average.

What is a twin prime?

A pair of primes two apart, such as 11 and 13, or 29 and 31. There are eight such pairs below 100. Whether there are infinitely many is one of the oldest unsolved problems in mathematics — it is widely believed to be true and has never been proved, though the gap between what is known and what is believed has narrowed considerably since 2013.

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.

Is 21 a Prime Number?

21 is a Fibonacci number that is also the product of two odd primes, 3 × 7, and one of the composites that looks prime at a glance because it is odd and does not end in 5. The digit-sum test settles it: 2 + 1 = 3. It is not prime. 21 = 3 × 7, so it has 4 divisors rather than two, and the smallest fa...

YOUR NUMBER

NUMBER

WORTH TRYING

21 IS

NOT PRIME

SMALLEST FACTOR

21 = 3 × 7

Prime factorisation

3 × 7

Number of divisors

Sum of divisors

Totient φ(n)

Previous prime

Next prime

FACTOR BREAKDOWN

PRIME

POWER

CONTRIBUTES

STEPS

Trial-divide 21 by every prime up to √21 ≈ 4 — beyond that a factor would have a partner below it

The smallest prime divisor is 3, so 21 is composite

Full factorisation: 3 × 7

21 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 21 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: 21 is not prime, because 21 = 3 × 7. Nothing needs typing — this page opens the calculator with 21 loaded.
2
Read the STEPS panel to see the reasoning: the smallest prime divisor of 21 is 3, which is enough to settle it, and the full factorisation 3 × 7 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 21 a Prime Number?

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HOW TO USE

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