Prime Number Identification

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number.

We need to examine each given number to determine if it fits the definition of a prime number.

Analyzing the Options

Checking 183

To check if 183 is prime, we can try dividing it by small prime numbers. The sum of the digits of 183 is $1 + 8 + 3 = 12$. Since 12 is divisible by 3, 183 is also divisible by 3.

We can confirm this division:

$$ \frac{183}{3} = 61 $$

Since 183 has a divisor (3) other than 1 and itself, 183 is a composite number.

Checking 121

We can try dividing 121 by small prime numbers. Let's start with 11.

$$ \frac{121}{11} = 11 $$

Since 121 can be expressed as $11 \times 11$, it has a divisor (11) other than 1 and itself. Thus, 121 is a composite number.

Checking 157

To check if 157 is prime, we need to test for divisibility by prime numbers up to the square root of 157. The square root of 157 is approximately $\sqrt{157} \approx 12.53$. The prime numbers less than or equal to 12 are 2, 3, 5, 7, and 11.

• Is 157 divisible by 2? No, because 157 is an odd number.
• Is 157 divisible by 3? The sum of digits is $1+5+7=13$. 13 is not divisible by 3, so 157 is not divisible by 3.
• Is 157 divisible by 5? No, because 157 does not end in 0 or 5.
• Is 157 divisible by 7? $157 \div 7 = 22$ with a remainder of $157 - 7 \times 22 = 157 - 154 = 3$. Not divisible by 7.
• Is 157 divisible by 11? $157 = 11 \times 14 + 3$. 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.

Checking 10201

Let's try to find factors for 10201. We can notice that this number is close to $100^2 = 10000$. Let's try the next integer, 101.

$$ 101 \times 101 = (100 + 1) \times (100 + 1) = 10000 + 100 + 100 + 1 = 10201 $$

Since 10201 can be expressed as $101 \times 101$, it has a divisor (101) other than 1 and itself. Thus, 10201 is a composite number.

Conclusion

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