What are the factors of 28?

28 has 6 divisors in total: 1, 2, 4, 7, 14, 28. Its prime factorisation is 2^2 × 7, 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 56.

What is the nearest prime to 28?

The prime immediately below 28 is 23, and the one immediately above is 29. Since 28 is composite it sits inside the run between them, 5 numbers wide.

What is Euler's totient of 28?

φ(28) = 12. The totient counts how many numbers below 28 share no common factor with it. Computing it from the factorisation 2^2 × 7 gives 28 × (1 − 1/2) × (1 − 1/7) = 12.

What is a perfect number?

A number equal to the sum of its own proper divisors. 6 is the first, since 1 + 2 + 3 = 6, and 28 is the second, since 1 + 2 + 4 + 7 + 14 = 28. They are extraordinarily rare: the next two are 496 and 8,128, and only about fifty are known in total. Every one found so far is even and pairs with a Mersenne prime.

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.

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.

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.

What makes a number prime?

Having exactly two divisors: 1 and itself. That is the entire definition, and it is why 1 does not qualify — it has only one divisor. Everything above 1 that is not prime is composite, and by the fundamental theorem of arithmetic every composite breaks into primes in exactly one way.

Is 28 a Prime Number?

28 is the second perfect number, equal to the sum of 1, 2, 4, 7 and 14. Perfect numbers are strikingly rare — the next is 496 — and every known one is even and pairs with a Mersenne prime, in this case 7. It is not prime. 28 = 2 × 2 × 7, so it has 6 divisors rather than two, and the smallest factor ...

YOUR NUMBER

NUMBER

WORTH TRYING

28 IS

NOT PRIME

SMALLEST FACTOR

28 = 2^2 × 7 · and a perfect number: its proper divisors sum to itself

Prime factorisation

2^2 × 7

Number of divisors

Sum of divisors

Totient φ(n)

Previous prime

Next prime

FACTOR BREAKDOWN

PRIME

POWER

CONTRIBUTES

STEPS

Trial-divide 28 by every prime up to √28 ≈ 5 — beyond that a factor would have a partner below it

The smallest prime divisor is 2, so 28 is composite

Full factorisation: 2^2 × 7

28 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 28 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: 28 is not prime, because 28 = 2 × 2 × 7. Nothing needs typing — this page opens the calculator with 28 loaded.
2
Read the STEPS panel to see the reasoning: the smallest prime divisor of 28 is 2, which is enough to settle it, and the full factorisation 2^2 × 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 29, 2026 · Formula verified · Deterministic Miller–Rabin, so a result of PRIME is proven rather than probable · Eagle-eyed accuracy for every calculation.

Is 28 a Prime Number?

YOUR NUMBER

HOW TO USE

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