What are the factors of 40?

40 has 8 divisors in total: 1, 2, 4, 5, 8, 10, 20, 40. Its prime factorisation is 2^3 × 5, and every one of those 8 divisors is built from those primes. The divisor count follows directly from the exponents: (3 + 1) × (1 + 1) = 8. They sum to 90.

What is the nearest prime to 40?

The prime immediately below 40 is 37, and the one immediately above is 41. Since 40 is composite it sits inside the run between them, 3 numbers wide.

What is Euler's totient of 40?

φ(40) = 16. The totient counts how many numbers below 40 share no common factor with it. Computing it from the factorisation 2^3 × 5 gives 40 × (1 − 1/2) × (1 − 1/5) = 16.

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.

What is the Sieve of Eratosthenes?

A method from around 240 BC and still essentially the fastest way to list every prime below a limit. Take the first uncrossed number and strike out all its multiples, then move to the next survivor and repeat. Each pass may start at the square of its prime, because everything smaller was already struck, and the passes stop once the prime exceeds the square root of the limit.

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.

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.

Are there patterns in where primes fall?

Every prime above 3 sits one away from a multiple of 6, which rules out four of every six numbers immediately. Beyond that the sequence is famously irregular: gaps vary unpredictably, and the deepest questions about their distribution, including the Riemann Hypothesis, remain open after more than a century and a half.

Is 40 a Prime Number?

40 is the product 2³ × 5, with eight divisors — the same count as 24 and 30, reached by a different arrangement of exponents. That three such different-looking numbers share a divisor count is the clearest illustration that the count depends on the exponents alone. It is not prime. 40 = 2 × 2 × 2 × ...

YOUR NUMBER

NUMBER

WORTH TRYING

40 IS

NOT PRIME

SMALLEST FACTOR

40 = 2^3 × 5

Prime factorisation

2^3 × 5

Number of divisors

Sum of divisors

Totient φ(n)

Previous prime

Next prime

FACTOR BREAKDOWN

PRIME

POWER

CONTRIBUTES

STEPS

Trial-divide 40 by every prime up to √40 ≈ 6 — beyond that a factor would have a partner below it

The smallest prime divisor is 2, so 40 is composite

Full factorisation: 2^3 × 5

40 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 40 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: 40 is not prime, because 40 = 2 × 2 × 2 × 5. Nothing needs typing — this page opens the calculator with 40 loaded.
2
Read the STEPS panel to see the reasoning: the smallest prime divisor of 40 is 2, which is enough to settle it, and the full factorisation 2^3 × 5 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 40 a Prime Number?

YOUR NUMBER

HOW TO USE

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