Understanding Prime Numbers

A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. Numbers that have more than two divisors are called composite numbers.

Analyzing the Options

We need to determine which of the given numbers is a prime number by checking their divisors.

• Option 1: 157
  To check if 157 is prime, we test divisibility by prime numbers up to $\sqrt{157}$. Since $\sqrt{157} \approx 12.53$, we only need to check primes 2, 3, 5, 7, and 11.
  • 157 is not divisible by 2 (it is odd).
  • The sum of digits $1+5+7=13$, which is not divisible by 3, so 157 is not divisible by 3.
  • 157 does not end in 0 or 5, so it is not divisible by 5.
  • $157 \div 7 \approx 22.43$ (remainder is 3). Not divisible by 7.
  • $157 \div 11 \approx 14.27$ (remainder is 3). Not divisible by 11.
  Since 157 is not divisible by any prime number less than or equal to its square root, 157 is a prime number.
• Option 2: 121
  $121 = 11 \times 11 = 11^2$. Since it has divisors other than 1 and itself (namely 11), 121 is not a prime number.
• Option 3: 1331
  $1331 = 11 \times 11 \times 11 = 11^3$. Since it has divisors other than 1 and itself (namely 11), 1331 is not a prime number.
• Option 4: 81
  $81 = 9 \times 9 = 9^2$ or $3 \times 3 \times 3 \times 3 = 3^4$. Since it has divisors other than 1 and itself (namely 3 and 9), 81 is not a prime number.

Conclusion

Based on the analysis, only 157 fits the definition of a prime number.