Identifying the Non-Prime Number

The question asks to identify which number from the given options is NOT a prime number. A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. A number that is not prime is called a composite number.

Method for Primality Testing

To determine if a number $n$ is prime, we can test its divisibility by prime numbers starting from 2 up to the square root of $n$ ($\sqrt{n}$). If $n$ is divisible by any prime number in this range, it is a composite number; otherwise, it is prime.

Analysis of Options

• Option 1: 811

  Calculate the square root: $\sqrt{811} \approx 28.47$. We need to check primes up to 23 (primes less than 28.47 are 2, 3, 5, 7, 11, 13, 17, 19, 23).

  • 811 is not divisible by 2 (it's odd).
  • Sum of digits $8+1+1=10$, not divisible by 3.
  • Does not end in 0 or 5, so not divisible by 5.
  • $811 \div 7 \approx 115.86$
  • $811 \div 11 \approx 73.73$
  • $811 \div 13 \approx 62.38$
  • $811 \div 17 \approx 47.71$
  • $811 \div 19 \approx 42.68$
  • $811 \div 23 \approx 35.26$

  Since 811 is not divisible by any prime number less than or equal to its square root, 811 is a prime number.

• Option 2: 317

  Calculate the square root: $\sqrt{317} \approx 17.80$. We need to check primes up to 17 (primes are 2, 3, 5, 7, 11, 13, 17).

  • 317 is not divisible by 2, 3 ($3+1+7=11$), or 5.
  • $317 \div 7 \approx 45.29$
  • $317 \div 11 \approx 28.82$
  • $317 \div 13 \approx 24.38$
  • $317 \div 17 \approx 18.65$

  Since 317 is not divisible by any prime number less than or equal to its square root, 317 is a prime number.

• Option 3: 817

  Calculate the square root: $\sqrt{817} \approx 28.58$. We need to check primes up to 23.

  • 817 is not divisible by 2, 3 ($8+1+7=16$), or 5.
  • $817 \div 7 \approx 116.71$
  • $817 \div 11 \approx 74.27$
  • $817 \div 13 \approx 62.85$
  • $817 \div 17 \approx 48.06$
  • $817 \div 19 = 43$

  Since 817 is divisible by 19 (and 43), 817 is NOT a prime number (it's a composite number).

• Option 4: 313

  Calculate the square root: $\sqrt{313} \approx 17.69$. We need to check primes up to 17.

  • 313 is not divisible by 2, 3 ($3+1+3=7$), or 5.
  • $313 \div 7 \approx 44.71$
  • $313 \div 11 \approx 28.45$
  • $313 \div 13 \approx 24.08$
  • $313 \div 17 \approx 18.41$

  Since 313 is not divisible by any prime number less than or equal to its square root, 313 is a prime number.

Conclusion

Based on the analysis, the number 817 is divisible by 19 and 43, meaning it has factors other than 1 and itself. Therefore, 817 is the number that is NOT prime.