Prime Number Checker

A prime number is a whole number greater than 1 with exactly two distinct divisors: 1 and itself. A composite number has more than two. So 7 is prime (only $1 \times 7$), while 9 is composite ($1 \times 9$ and $3 \times 3$).

1 is not prime. It has only one divisor, not two. That is not a technicality invented to annoy students: if 1 counted as prime, every number would have endlessly many prime factorisations ($6 = 2 \times 3 = 1 \times 2 \times 3 = 1 \times 1 \times 2 \times 3$), and the guarantee that each number factors into primes in exactly one way would collapse.

2 is prime — and it is the only even prime, because every other even number has 2 as a third divisor. Students often expect 2 to be the exception in the other direction; it is not.

The first primes are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47$.

To check whether $n$ is prime, try dividing by each prime from 2 upward, and stop once the prime exceeds $\sqrt{n}$.

Why stop there? If $n = a \times b$ with both factors above $\sqrt{n}$, then $a \times b > n$ — impossible. So any composite number must have a factor at or below its square root. For $n = 97$, $\sqrt{97} \approx 9.85$, so testing 2, 3, 5 and 7 settles it completely.

The divisibility shortcuts make this fast without a calculator:

One clean division with no remainder is enough to declare the number composite — you can stop immediately.

• 51 looks prime but $5 + 1 = 6$ is a multiple of 3, so $51 = 3 \times 17$.
• 91 slips past the 2, 3 and 5 tests, then fails at 7: $91 = 7 \times 13$.
• 361 is $19^2$, so it is composite even though nothing small divides it. Always test all the way to $\sqrt{n}$ before declaring a number prime.
• 9 is not prime. Odd does not mean prime; $9 = 3 \times 3$.
• Stopping too early. Checking only 2, 3 and 5 on a three-digit number proves nothing. For $n = 223$ you must reach 13, since $\sqrt{223} \approx 14.9$.
• Confusing this with prime factorisation. "Is 113 prime?" asks for yes or no; "prime factors of 113" asks for the full breakdown — which here is just $113$ itself, because 113 is prime.