Understanding Composite Numbers

A composite number is a positive integer that has at least one divisor other than 1 and itself. In simpler terms, it can be formed by multiplying two smaller positive integers. For example, 6 is a composite number because $6 = 2 \times 3$.

In contrast, a prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. For example, 7 is a prime number because its only divisors are 1 and 7.

The question asks us to identify which of the given numbers (241, 261, 271, 251) is composite.

How to Determine if a Number is Composite

To check if a number is composite, we can test its divisibility by prime numbers starting from the smallest primes (2, 3, 5, 7, 11, etc.). We only need to check for prime divisors up to the square root of the number. If we find any prime number that divides the given number evenly (with no remainder) within this range, the number is composite. If no such prime divisor is found up to its square root, the number is prime.

Analyzing Each Option

Let's examine each given number to determine if it is composite or prime.

Checking the Number 241

First, estimate the square root of 241. $\sqrt{241} \approx 15.5$. We need to check for prime divisors up to 15.5. The prime numbers to test are 2, 3, 5, 7, 11, and 13.

• Is 241 divisible by 2? No, because 241 is an odd number.
• Is 241 divisible by 3? Sum of digits = $2+4+1 = 7$. 7 is not divisible by 3, so 241 is not divisible by 3.
• Is 241 divisible by 5? No, because it does not end in 0 or 5.
• Is 241 divisible by 7? $241 \div 7 = 34$ with a remainder of $3$. No.
• Is 241 divisible by 11? $241 \div 11 = 21$ with a remainder of $10$. No.
• Is 241 divisible by 13? $241 \div 13 = 18$ with a remainder of $7$. No.

Since 241 is not divisible by any prime number up to its square root, 241 appears to be a prime number.

Checking the Number 261

The square root of 261 is $\sqrt{261} \approx 16.1$. We need to check for prime divisors up to 16.1. The prime numbers to test are 2, 3, 5, 7, 11, and 13.

• Is 261 divisible by 2? No, because 261 is an odd number.
• Is 261 divisible by 3? Sum of digits = $2+6+1 = 9$. 9 is divisible by 3 ($9 \div 3 = 3$), so 261 is divisible by 3.

Since 261 is divisible by 3 ($261 \div 3 = 87$), it has a divisor (3) other than 1 and itself. Therefore, 261 is a composite number.

We can stop here, as we have found that 261 is composite. However, for completeness, let's look at the other options.

Checking the Number 271

The square root of 271 is $\sqrt{271} \approx 16.4$. We need to check for prime divisors up to 16.4. The prime numbers to test are 2, 3, 5, 7, 11, and 13.

• Is 271 divisible by 2? No, odd number.
• Is 271 divisible by 3? Sum of digits = $2+7+1 = 10$. Not divisible by 3.
• Is 271 divisible by 5? No, doesn't end in 0 or 5.
• Is 271 divisible by 7? $271 \div 7 = 38$ with a remainder of $5$. No.
• Is 271 divisible by 11? $271 \div 11 = 24$ with a remainder of $7$. No.
• Is 271 divisible by 13? $271 \div 13 = 20$ with a remainder of $11$. No.

Since 271 is not divisible by any prime number up to its square root, 271 appears to be a prime number.

Checking the Number 251

The square root of 251 is $\sqrt{251} \approx 15.8$. We need to check for prime divisors up to 15.8. The prime numbers to test are 2, 3, 5, 7, 11, and 13.

• Is 251 divisible by 2? No, odd number.
• Is 251 divisible by 3? Sum of digits = $2+5+1 = 8$. Not divisible by 3.
• Is 251 divisible by 5? No, doesn't end in 0 or 5.
• Is 251 divisible by 7? $251 \div 7 = 35$ with a remainder of $6$. No.
• Is 251 divisible by 11? $251 \div 11 = 22$ with a remainder of $9$. No.
• Is 251 divisible by 13? $251 \div 13 = 19$ with a remainder of $4$. No.

Since 251 is not divisible by any prime number up to its square root, 251 appears to be a prime number.

Summary of Findings

From the analysis, only the number 261 is composite.

Revision Table: Composite Numbers

Additional Information: Divisibility Rules and Number Properties

Using divisibility rules can make it faster to check if a number is composite.

• Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
• Divisibility by 7: Double the last digit and subtract it from the number formed by the remaining digits. If the result is divisible by 7, the original number is divisible by 7. Repeat if necessary. (e.g., for 241: $24 - 2 \times 1 = 22$, not div by 7).
• Divisibility by 11: Alternately add and subtract the digits of the number. If the result is divisible by 11 (including 0), the number is divisible by 11. (e.g., for 261: $1 - 6 + 2 = -3$, not div by 11. Wait, for 261: $2-6+1 = -3$, still not div by 11. Let's check $261 \div 11 = 23$ with remainder 8. My earlier check was wrong. The sum of alternating digits rule confirms 261 is not divisible by 11, but we already know it's divisible by 3).

These rules are helpful shortcuts when testing small prime divisors.