Understanding Composite and Prime Numbers

In number theory, positive integers greater than 1 are classified as either prime or composite. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A composite number is a positive integer that has at least one divisor other than 1 and itself. This means a composite number can be formed by multiplying two smaller positive integers.

The question asks us to identify which of the given numbers is not composite. This is equivalent to asking which of the given numbers is a prime number.

Testing Each Number for Compositeness

To determine if a number is composite or prime, we can test for divisibility by prime numbers starting from 2. If we find any divisor other than 1 and the number itself, the number is composite. If we test all prime numbers up to the square root of the number and find no divisors, then the number is prime.

Analyzing Option 1: 203

Let's test 203 for small prime divisors:

• Is it divisible by 2? No, it's an odd number.
• Is it divisible by 3? The sum of digits is $2+0+3 = 5$, which is not divisible by 3. So, 203 is not divisible by 3.
• Is it divisible by 5? No, it does not end in 0 or 5.
• Is it divisible by 7? Let's divide: $203 \div 7$. $7 \times 20 = 140$, $203 - 140 = 63$. $7 \times 9 = 63$. So, $203 = 7 \times 20 + 7 \times 9 = 7 \times (20+9) = 7 \times 29$.

Since 203 can be expressed as the product of two smaller integers (7 and 29), 203 is a composite number.

Analyzing Option 2: 109

Let's test 109 for small prime divisors. We need to test primes up to $\sqrt{109}$. Since $10^2 = 100$ and $11^2 = 121$, $\sqrt{109}$ is between 10 and 11. The primes we need to test are 2, 3, 5, and 7.

• Is it divisible by 2? No, it's odd.
• Is it divisible by 3? The sum of digits is $1+0+9 = 10$, which is not divisible by 3. So, 109 is not divisible by 3.
• Is it divisible by 5? No, it does not end in 0 or 5.
• Is it divisible by 7? Let's divide: $109 \div 7$. $7 \times 10 = 70$, $109 - 70 = 39$. $7 \times 5 = 35$. So, $109 = 7 \times 15 + 4$. Not divisible by 7.

Since 109 is not divisible by any prime less than or equal to its square root, 109 has no positive divisors other than 1 and itself. Therefore, 109 is a prime number. A prime number is not composite.

Analyzing Option 3: 209

Let's test 209 for small prime divisors:

• Not divisible by 2 (odd).
• Not divisible by 3 ($2+0+9 = 11$).
• Not divisible by 5 (does not end in 0 or 5).
• Is it divisible by 7? $209 = 7 \times 29 + 6$. Not divisible by 7.
• Is it divisible by 11? $209 \div 11$. $11 \times 10 = 110$, $209 - 110 = 99$. $11 \times 9 = 99$. So, $209 = 11 \times 10 + 11 \times 9 = 11 \times (10+9) = 11 \times 19$.

Since 209 can be expressed as the product of two smaller integers (11 and 19), 209 is a composite number.

Analyzing Option 4: 161

Let's test 161 for small prime divisors:

• Not divisible by 2 (odd).
• Not divisible by 3 ($1+6+1 = 8$).
• Not divisible by 5 (does not end in 0 or 5).
• Is it divisible by 7? $161 \div 7$. $7 \times 20 = 140$, $161 - 140 = 21$. $7 \times 3 = 21$. So, $161 = 7 \times 20 + 7 \times 3 = 7 \times (20+3) = 7 \times 23$.

Since 161 can be expressed as the product of two smaller integers (7 and 23), 161 is a composite number.

Conclusion: Identifying the Non-Composite Number

Based on our analysis:

• 203 is composite ($7 \times 29$).
• 109 is prime (only divisors are 1 and 109).
• 209 is composite ($11 \times 19$).
• 161 is composite ($7 \times 23$).

The number that is not composite is 109.

Revision Table: Prime vs. Composite Summary

Additional Information on Prime and Composite Numbers

Understanding the difference between prime and composite numbers is fundamental in number theory. Here are some key points:

• The number 1 is neither prime nor composite. It is a special case.
• The smallest prime number is 2. It is the only even prime number.
• Every composite number can be written as a unique product of prime numbers (this is the Fundamental Theorem of Arithmetic). For example, $12 = 2^2 \times 3$.
• To check if a number $N$ is prime, you only need to test for divisibility by prime numbers up to $\sqrt{N}$. If none of these primes divide $N$, then $N$ is prime. This significantly speeds up the process for larger numbers.