What are the factors of 36?

36 has 9 divisors in total: 1, 2, 3, 4, 6, 9, 12, 18, 36. Its prime factorisation is 2^2 × 3^2, and every one of those 9 divisors is built from those primes. The divisor count follows directly from the exponents: (2 + 1) × (2 + 1) = 9. They sum to 91.

What is the nearest prime to 36?

The prime immediately below 36 is 31, and the one immediately above is 37. Since 36 is composite it sits inside the run between them, 5 numbers wide.

What is Euler's totient of 36?

φ(36) = 12. The totient counts how many numbers below 36 share no common factor with it. Computing it from the factorisation 2^2 × 3^2 gives 36 × (1 − 1/2) × (1 − 1/3) = 12.

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.

What are the quick divisibility tests?

A number is divisible by 2 if it ends in an even digit, by 3 if its digits sum to a multiple of 3, by 5 if it ends in 0 or 5, and by 11 if the alternating sum of its digits is a multiple of 11. Those four catch most small composites instantly. What they miss are numbers whose smallest factor is 7 or larger, which is exactly where people go wrong.

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 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.

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.

Is 36 a Prime Number?

36 is the smallest number with nine divisors, and the square of 6 as well as 2² × 3². Squares always have an odd number of divisors, because divisors pair off except for the square root, which pairs with itself. It is not prime. 36 = 2 × 2 × 3 × 3, so it has 9 divisors rather than two, and the small...

YOUR NUMBER

NUMBER

WORTH TRYING

36 IS

NOT PRIME

SMALLEST FACTOR

36 = 2^2 × 3^2

Prime factorisation

2^2 × 3^2

Number of divisors

Sum of divisors

Totient φ(n)

Previous prime

Next prime

FACTOR BREAKDOWN

PRIME

POWER

CONTRIBUTES

STEPS

Trial-divide 36 by every prime up to √36 ≈ 6 — beyond that a factor would have a partner below it

The smallest prime divisor is 2, so 36 is composite

Full factorisation: 2^2 × 3^2

36 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 36 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: 36 is not prime, because 36 = 2 × 2 × 3 × 3. Nothing needs typing — this page opens the calculator with 36 loaded.
2
Read the STEPS panel to see the reasoning: the smallest prime divisor of 36 is 2, which is enough to settle it, and the full factorisation 2^2 × 3^2 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 36 a Prime Number?

YOUR NUMBER

HOW TO USE

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