Understanding Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In simpler terms, a prime number is only divisible by 1 and the number itself.

Numbers greater than 1 that are not prime numbers are called composite numbers. Composite numbers have more than two factors (including 1 and the number itself).

Checking Each Option for Primality

Let's examine each number provided in the options to determine if it is a prime number by finding its factors.

• Option 1: 15

  The factors of 15 are the numbers that divide into 15 without leaving a remainder. The factors of 15 are 1, 3, 5, and 15.

  Since 15 has factors other than 1 and itself (specifically 3 and 5), it is not a prime number. It is a composite number.

• Option 2: 23

  The factors of 23 are the numbers that divide into 23 without leaving a remainder. Let's check small numbers.

  • 23 is not divisible by 2 (it's odd).
  • The sum of digits (2+3=5) is not divisible by 3, so 23 is not divisible by 3.
  • 23 does not end in 0 or 5, so it's not divisible by 5.
  • We only need to check for prime factors up to the square root of 23, which is about 4.8. The primes to check are 2 and 3 (5 is too large). We already checked 2 and 3.

  The only positive divisors of 23 are 1 and 23. Therefore, 23 fits the definition of a prime number.

• Option 3: 12

  The factors of 12 are 1, 2, 3, 4, 6, and 12.

  Since 12 has factors other than 1 and itself (like 2, 3, 4, and 6), it is not a prime number. It is a composite number.

• Option 4: 75

  The factors of 75 include 1, 3, 5, 15, 25, and 75.

  Since 75 has factors other than 1 and itself (like 3, 5, 15, 25), it is not a prime number. It is a composite number.

Conclusion

Based on the analysis of the factors, only 23 meets the criteria of being a prime number.