Why is 1 not a prime number?

A prime must have exactly two distinct positive factors: 1 and itself. The number 1 has only one factor (just 1), so it does not meet the definition. Excluding 1 also keeps the Fundamental Theorem of Arithmetic working correctly — every integer greater than 1 has a unique prime factorization.

List of Prime Numbers — Definition, Formula & Examples

A list of prime numbers is a reference table of whole numbers greater than 1 that have exactly two factors: 1 and themselves. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

A prime number $p$ is an integer greater than 1 whose only positive divisors are 1 and $p$ . The set of all primes is infinite, as proven by Euclid, and begins $\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, \ldots\}$ . Every integer greater than 1 that is not prime is called composite.

Key Formula

\text{Test divisors: only primes } p \leq \sqrt{n}

Where:

$n$ = The number you are testing for primality
$p$ = Each prime candidate divisor up to the square root of n

How It Works

To check whether a number

n

is prime, test whether any prime up to

\sqrt{n}

divides it evenly. If none do, the number is prime. For example, to test 37, you only need to check primes up to

\sqrt{37} \approx 6.1

, which means checking 2, 3, and 5. Since none of them divide 37 evenly, 37 is prime. Having a reference list saves time when you need primes for factoring, finding GCFs, or simplifying fractions. **Primes from 2 to 100:** 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 (25 primes). **Primes from 101 to 500:** 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499. **Primes from 501 to 1000:** 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997. There are 168 primes up to 1000.

Worked Example

Problem: Is 91 a prime number?

Step 1: Find the square root of 91 to determine which primes to test.

\sqrt{91} \approx 9.5

Step 2: List the primes up to 9.5: 2, 3, 5, and 7.

Step 3: Test each: 91 is odd (not divisible by 2), digit sum is 10 (not divisible by 3), doesn't end in 0 or 5 (not divisible by 5). Now try 7.

91 \div 7 = 13

Step 4: Since 7 divides 91 evenly, 91 is not prime — it is composite.

91 = 7 \times 13

Answer: 91 is not prime. Its prime factorization is

7 \times 13

Another Example

Problem: Find all prime numbers between 40 and 60.

Step 1: List the candidates: 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59.

Step 2: Eliminate even numbers (divisible by 2): removes 42, 44, 46, 48, 50, 52, 54, 56, 58.

Step 3: Eliminate multiples of 3 (digit-sum test): 45 (4+5=9), 51 (5+1=6), 57 (5+7=12). Eliminate multiples of 5: 55. Eliminate multiples of 7: 49 = 7×7.

Step 4: The surviving numbers are the primes between 40 and 60.

Answer: The primes between 40 and 60 are 41, 43, 47, 53, and 59.

Visualization

Why It Matters

You will use prime numbers constantly in middle-school and pre-algebra when simplifying fractions, finding GCFs and LCMs, and building factor trees. In computer science and cybersecurity, large primes are the foundation of RSA encryption, which secures online banking and messaging. Memorizing or having quick access to the primes up to 100 will speed up nearly every factoring problem you encounter.

Common Mistakes

Mistake: Calling 1 a prime number.

Correction: 1 has only one positive factor, not two. By definition, primes must have exactly two distinct positive divisors.

Mistake: Assuming all odd numbers are prime (e.g., treating 9, 15, 21, 49, or 91 as prime).

Correction: Being odd is necessary (except for 2) but not sufficient. Always test divisibility by small primes up to the square root of the number.

Related Terms

Prime Factorization — Breaking a number into its prime factors
Factor Tree — Visual method for finding prime factors
Fundamental Theorem of Arithmetic — Every integer has a unique prime factorization
Factor of an Integer — Primes are numbers with exactly two factors
Greatest Common Factor — Found using shared prime factors
Least Common Multiple — Computed from prime factorizations
Relatively Prime — Two numbers sharing no prime factors
Prime Divisor — A prime that divides a given number