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Prime Number

Prime Number

A positive integer which has only 1 and the number itself as factors. For example, 2, 3, 5, 7, 11, 13, etc. are all primes. By convention, the number 1 is not prime.

 

 

See also

Composite number

Worked Example

Problem: Determine whether 29 is a prime number.
Step 1: To check if 29 is prime, you only need to test divisors up to the square root of 29. Since the square root of 29 is between 5 and 6, you test the primes 2, 3, and 5.
29≈5.39\sqrt{29} \approx 5.39
Step 2: Check divisibility by 2. Since 29 is odd, it is not divisible by 2.
29÷2=14.529 \div 2 = 14.5
Step 3: Check divisibility by 3. The digits of 29 sum to 11, which is not divisible by 3, so 29 is not divisible by 3.
29÷3=9.6‾29 \div 3 = 9.\overline{6}
Step 4: Check divisibility by 5. Since 29 does not end in 0 or 5, it is not divisible by 5.
29÷5=5.829 \div 5 = 5.8
Step 5: No prime up to the square root of 29 divides it evenly, so 29 has no factors other than 1 and 29.
Answer: 29 is a prime number.

Another Example

Problem: Determine whether 51 is a prime number.
Step 1: Find the square root of 51 to determine which primes to test.
51≈7.14\sqrt{51} \approx 7.14
Step 2: Check divisibility by 2. Since 51 is odd, it is not divisible by 2.
Step 3: Check divisibility by 3. The digits of 51 sum to 6, and 6 is divisible by 3, so 51 is divisible by 3.
51÷3=1751 \div 3 = 17
Step 4: Since 51 can be written as 3 × 17, it has factors other than 1 and itself.
51=3×1751 = 3 \times 17
Answer: 51 is NOT a prime number — it is composite.

Frequently Asked Questions

Why is 1 not a prime number?
A prime number must have exactly two distinct positive factors: 1 and itself. The number 1 has only one factor (itself), so it does not meet this requirement. Excluding 1 from the primes also keeps the Fundamental Theorem of Arithmetic clean: every integer greater than 1 has a unique prime factorization.
Is 2 a prime number, and why is it the only even prime?
Yes, 2 is prime because its only positive factors are 1 and 2. Every other even number is divisible by 2, which gives it a factor besides 1 and itself. That makes 2 the one and only even prime number.

Prime Number vs. Composite Number

A prime number has exactly two distinct positive factors (1 and itself), while a composite number has three or more positive factors. For example, 7 is prime (factors: 1, 7), but 12 is composite (factors: 1, 2, 3, 4, 6, 12). The number 1 is neither prime nor composite — it stands alone as a special case.

Why It Matters

Prime numbers are the building blocks of all positive integers. The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed as a unique product of primes. Beyond pure mathematics, primes are essential in modern cryptography — encryption methods like RSA rely on the difficulty of factoring large numbers into their prime components to keep digital communications secure.

Common Mistakes

Mistake: Thinking 1 is a prime number.
Correction: A prime must have exactly two distinct positive factors. The number 1 has only one factor (itself), so by definition it is not prime.
Mistake: Assuming all odd numbers are prime.
Correction: Many odd numbers are composite. For example, 9 = 3 × 3, 15 = 3 × 5, and 21 = 3 × 7 are all odd but not prime. Always check for factors before concluding a number is prime.

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