Enter one number and see everything about it at once — prime factorisation, every divisor, whether it is prime, its representation in other bases, as a Roman numeral, and written out in words. Most tools answer one of these; this answers all of them from a single entry.
Number Properties Calculator
LiveWhat each property means
| Property | What it tells you |
|---|---|
| Prime factorisation | The unique set of primes that multiply to your number. By the fundamental theorem of arithmetic, every whole number above 1 has exactly one such set — no exceptions, no alternatives |
| Divisor count | Read straight off the factorisation: add one to each exponent and multiply. 360 = 2³ × 3² × 5¹ gives (3+1)(2+1)(1+1) = 24 divisors, without listing any of them |
| Euler's totient φ(n) | How many numbers below n share no common factor with it. Central to RSA encryption, which is why a number-theory curiosity from 1763 secures your bank transfers |
| Digital root | Sum the digits repeatedly until one digit remains. Equals n mod 9, which is what makes casting out nines work as an arithmetic check |
| Perfect number | Equals the sum of its proper divisors. 6, 28, 496, 8128 — only 51 are known, and whether any odd one exists is unsolved after 2,000 years |
Why prime factorisation matters beyond homework
It looks like a school exercise and underpins most of modern cryptography. The asymmetry is the whole point: multiplying two large primes is instant; recovering them from the product is not.
Multiplying two 300-digit primes takes a computer microseconds. Factoring the 600-digit result back into those primes would take the fastest known algorithms longer than the age of the universe. RSA encryption is built entirely on that gap — your public key is the product, and your private key depends on knowing the factors.
The practical uses are more mundane and just as real. Factorisation gives you the lowest common denominator for adding fractions, the greatest common divisor for simplifying them, and the answer to any question about arranging things into equal groups — which is why divisor counts turn up constantly in scheduling and packing problems.
Some numbers worth knowing
| Number | Why it is interesting | Detail |
|---|---|---|
| 360 | 24 divisors | The reason a circle has 360 degrees — it divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180 |
| 28 | Perfect | 1 + 2 + 4 + 7 + 14 = 28. The second perfect number, after 6 |
| 1729 | Taxicab number | The smallest expressible as a sum of two cubes in two ways: 1³+12³ and 9³+10³. Ramanujan's answer to Hardy's "rather dull" cab number |
| 2,147,483,647 | Mersenne prime | 2³¹ − 1, and the maximum value of a signed 32-bit integer — the cause of a great many software bugs |
| 7 | Awkward divisor | Coprime to 10, so 1/7 gives the repeating 0.142857… — and why 7-day weeks never align neatly with months |
Roman numerals, and their limits
| Symbol | Value | Subtractive pairs |
|---|---|---|
| I V X | 1, 5, 10 | IV = 4, IX = 9 |
| L C | 50, 100 | XL = 40, XC = 90 |
| D M | 500, 1000 | CD = 400, CM = 900 |
Two structural limits explain why the system was abandoned. There is no zero, and no place value — so arithmetic requires a physical abacus rather than pen and paper. And 3,999 is the practical ceiling without overbars, since there is no symbol above M. Try to write 4,000 and you need MMMM, which the subtractive convention was meant to avoid.
The calculator uses standard subtractive notation. Historical inscriptions are inconsistent — IIII appears on many clock faces to this day, including Big Ben's.
Common mistakes
Frequently asked questions
How do I find the prime factorisation of a number?
Divide by the smallest prime that goes in evenly, then repeat on the result. For 360: divide by 2 three times to reach 45, by 3 twice to reach 5, and 5 is prime — giving 2³ × 3² × 5. You only need to test primes up to the square root, because any factor above it pairs with one below.
How can I tell whether a number is prime?
Test divisibility by every prime up to its square root. For 97, that means 2, 3, 5 and 7 — none divide it, so 97 is prime. The square root shortcut is what makes this practical: checking 97 needs four tests rather than ninety-five. For very large numbers, probabilistic tests like Miller-Rabin are used instead.
Why is 1 not a prime number?
Because a prime has exactly two distinct positive divisors, and 1 has only itself. The deeper reason is that including it would break the fundamental theorem of arithmetic — every number's factorisation would stop being unique, since you could multiply in any number of 1s. Excluding it keeps the theorem clean, which is worth more than the convenience of a broader definition.
How many divisors does a number have?
Take the prime factorisation, add one to each exponent, and multiply the results. For 360 = 2³ × 3² × 5¹: (3+1)(2+1)(1+1) = 24. This works without listing a single divisor, which is why it is the standard method for large numbers where enumeration would be impractical.
What is a perfect number?
One that equals the sum of its proper divisors. 6 = 1+2+3, and 28 = 1+2+4+7+14. Only 51 are known, all even, and each corresponds to a Mersenne prime. Whether an odd perfect number exists is one of the oldest unsolved problems in mathematics — open since Euclid, roughly 2,300 years.
What is Euler's totient used for?
It counts the numbers below n that share no factor with it, and it sits at the centre of RSA encryption. Generating an RSA key pair requires φ(pq) for two large primes p and q — which is easy if you know the primes and effectively impossible if you only know their product. That gap is what secures the connection you are reading this over.
Why can't Roman numerals go above 3,999?
Because M is the largest symbol, so 4,000 would need MMMM — which the subtractive convention exists to avoid. Historically a bar over a numeral multiplied it by a thousand, but that is not part of standard modern notation. The absence of zero and of place value is the deeper limitation, and the reason the system could not survive contact with written arithmetic.
Is my input stored?
No. Everything runs in your browser with no server request, and works offline once the page has loaded.
Related calculators
See the full list of Math calculators, or try:
- Factor Calculator — factors and primality on their own
- GCF & LCM Calculator — factorisation applied to two numbers
- Roman Numeral Converter — both directions, with larger ranges
- Binary & Hex Converter — any base to any base
- Scientific Notation Calculator — very large and very small numbers
- Fraction Calculator — where common denominators come from