🔍 Prime Number Checker
Check whether a number is prime, and get its prime factorization plus the nearest primes above and below it.
Quick answer: 97 is prime: no whole number from 2 to 9 (√97 ≈ 9.8489) divides it evenly. By contrast, 91 is not prime because 91 = 7 × 13.
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Prime Number Checker inputs
Result
Is it prime?
97 is a prime number
Its only divisors are 1 and 97.
- Smallest prime factor
- 97
- Prime factorization
- 97
- Number of divisors
- 2
- Previous prime
- 89
- Next prime
- 101
Step by step
- √97 ≈ 9.8489, so only divisors up to 9 need testing
- No whole number from 2 to 9 divides 97 evenly, so it is prime
What makes a number prime?
A prime number is a whole number greater than 1 whose only divisors are 1 and itself. Numbers with more divisors are composite. The numbers 0 and 1 are neither.
n is prime ⇔ n > 1 and no integer d with 2 ≤ d ≤ √n divides n
Why only test up to the square root?
If n = a × b, then a and b can’t both be bigger than √n – their product would be too large. So any composite number has a factor no bigger than its square root, and you can stop searching there.
Worked examples
- 97: √97 ≈ 9.85. Test 2, 3, 5 and 7 – none divide 97, so it is prime.
- 91: √91 ≈ 9.54. 91 ÷ 7 = 13 exactly, so 91 = 7 × 13 is composite. It is a classic “looks prime” trap.
- 360: divide out 2 three times, 3 twice and 5 once: 360 = 2³ × 3² × 5, with (3 + 1)(2 + 1)(1 + 1) = 24 divisors.
The first primes
| Range | Primes |
|---|---|
| 1–25 | 2, 3, 5, 7, 11, 13, 17, 19, 23 |
| 26–50 | 29, 31, 37, 41, 43, 47 |
| 51–100 | 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 |
There are 25 primes below 100 and infinitely many in total, as Euclid proved over 2,000 years ago.
Quick divisibility tests
- 2: the last digit is even.
- 3: the digit sum is divisible by 3.
- 5: the number ends in 0 or 5.
- 11: the alternating digit sum is divisible by 11.
The checker handles every whole number up to one trillion exactly, using trial division by 2, 3 and numbers of the form 6k ± 1.
Frequently asked questions
Is 1 a prime number?
No. A prime has exactly two distinct divisors, 1 and itself, and 1 has only one. Excluding 1 also keeps prime factorizations unique – otherwise you could write 6 = 2 × 3 × 1 × 1 × …
Is 2 the only even prime?
Yes. Every other even number is divisible by 2, so it has at least three divisors (1, 2 and itself). That makes 2 the smallest prime and the only even one.
How do you check if a large number is prime?
Test divisibility by primes up to its square root – if none divide it, it is prime. For 1,000,000,007 that means checking divisors up to about 31,623, which the checker does in a fraction of a second.
What is prime factorization?
Writing a number as a product of primes, such as 360 = 2³ × 3² × 5. Every whole number above 1 has exactly one prime factorization (the fundamental theorem of arithmetic).