Details
- Factor pair: —
Step-by-step
Key steps used to check primality:
Prime Number Checker
This tool checks whether an integer is prime or composite and explains the divisibility checks used to decide. It tests only the divisors that matter up to $\sqrt{n}$. If the number is composite, it also shows the smallest divisor so you can immediately see a factor pair. Prime checking is useful for factoring, simplifying fractions, and many algebra routines. For more factor and divisibility tools, explore Number Theory.
Check whether an integer is prime.
Integer values only. Decimals are not allowed.
What Is a Prime Number?
A prime number is a positive integer greater than $1$ that has exactly two positive divisors: $1$ and itself. Numbers greater than $1$ that have additional divisors are called composite. The number $1$ is neither prime nor composite.
How Prime Checking Works
The fastest basic method for a single number is divisibility testing. If $n$ has a divisor other than $1$ and itself, it must have one that is at most $\sqrt{n}$. That means you only need to check potential divisors up to $\sqrt{n}$.
Why the $\sqrt{n}$ Limit Is Enough
If $n$ is composite, you can write it as $n = a·b$ with $1$ < $a$ ≤ $b$ < $n$. At least one of $a$ or $b$ must be ≤ $\sqrt{n}$, otherwise their product would exceed $n$. So if you find no divisor ≤ $\sqrt{n}$, $n$ is prime.
If you are simplifying ratios or reducing fractions, the GCD Calculator helps you remove common factors. When you need the smallest common denominator for fractions, the LCM Calculator is the right tool.
Common Edge Cases
- $n$ ≤ $1$: not prime.
- $n$ = $2$ or $n$ = $3$: prime.
- Even numbers greater than $2$: composite.
- Negative integers: not prime (primes are defined over positive integers).
What the Results Mean
If the tool returns Prime, it means no divisor was found after testing only the values that matter up to the square root of $n$. For a prime number, the only positive divisors are $1$ and itself.
If the tool returns Composite, it will also show the smallest divisor. That divisor is the first number greater than $1$ that divides $n$ exactly, and it immediately gives you a factor pair. This is useful when you need to factor numbers, reduce fractions, or simplify algebraic expressions.
If $n$ is $0$, $1$, or negative, the tool will mark it as not prime because primes are defined only for positive integers greater than $1$.
Worked Examples
Example 1: A prime number
Check $29$. Since it is not even and not divisible by $3$, test the next candidates up to $\sqrt{29}$ (which is a bit more than $5$). The only remaining divisor to test is $5$, and $29$ is not divisible by $5$.
Conclusion: $29$ is prime.
Example 2: A composite number
Check $91$. It is not even, and $9+1=10$ so it is not divisible by $3$. Try $5$ (no), then $7$: $91$ ÷ $7$ = $13$, so $7$ divides $91$.
$$91 = 7 \cdot 13$$ Conclusion: $91$ is composite, and the smallest divisor is $7$.
Example 3: Not prime by definition
Check $1$. Prime numbers must be greater than $1$, so $1$ is not prime. It is also not composite because it does not have two different positive divisors.
Conclusion: $1$ is neither prime nor composite.
Keep exploring Mathematics or jump back to Calculators to browse more tools.
Questions About the Prime Checker
Quick answers about primes, composites, and divisibility.
A prime number is a positive integer greater than $1$ that has exactly two positive divisors: $1$ and itself.
No. The number $1$ has only one positive divisor ($1$), so it is neither prime nor composite.
A composite number is a positive integer greater than $1$ that has more than two positive divisors.
You only need to test divisors up to $\sqrt{n}$. If no divisor $\le \sqrt{n}$ divides $n$, then $n$ is prime.
For a composite number, the smallest divisor is the smallest integer greater than $1$ that divides the number exactly.
No. Prime numbers are defined only for positive integers greater than $1$.
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