Understanding Prime Numbers

A prime number is a natural number greater than 1 that has only two divisors: 1 and itself. A number that has more than two divisors is called a composite number (or non-prime number).

Analyzing the Options

We need to check each number to see if it has divisors other than 1 and itself.

• 313: To check if 313 is prime, we test divisibility by primes up to $\sqrt{313} \approx 17.7$. Primes to check are 2, 3, 5, 7, 11, 13, 17.
  • 313 is not divisible by 2 (it's odd).
  • $3+1+3=7$, not divisible by 3.
  • Does not end in 0 or 5.
  • $313 \div 7 \approx 44.7$
  • $313 \div 11 \approx 28.45$
  • $313 \div 13 \approx 24.08$
  • $313 \div 17 \approx 18.41$
  Since 313 is not divisible by any prime number less than or equal to its square root, 313 is a prime number.
• 231:
  • Check divisibility by 3: The sum of the digits is $2+3+1=6$. Since 6 is divisible by 3, 231 is divisible by 3.
  • Calculation: $231 \div 3 = 77$.
  Because 231 has divisors 1, 3, 7, 11, 21, 33, 77, and 231, it is NOT a prime number.
• 241: Test divisibility by primes up to $\sqrt{241} \approx 15.5$. Primes: 2, 3, 5, 7, 11, 13.
  • 241 is not divisible by 2, 3 ($2+4+1=7$), or 5.
  • $241 \div 7 \approx 34.4$
  • $241 \div 11 \approx 21.9$
  • $241 \div 13 \approx 18.5$
  241 is a prime number.
• 211: Test divisibility by primes up to $\sqrt{211} \approx 14.5$. Primes: 2, 3, 5, 7, 11, 13.
  • 211 is not divisible by 2, 3 ($2+1+1=4$), or 5.
  • $211 \div 7 \approx 30.14$
  • $211 \div 11 \approx 19.18$
  • $211 \div 13 \approx 16.23$
  211 is a prime number.

Conclusion

The number 231 is divisible by 3 (and 7 and 11), meaning it has factors other than 1 and itself. Therefore, 231 is NOT a prime number.