Understanding Composite Numbers

A composite number is a positive integer that has at least one divisor other than 1 and itself. In simpler terms, if a number can be divided evenly by any number other than 1 and itself, it is a composite number. Numbers that only have two distinct positive divisors: 1 and the number itself, are called prime numbers. The number 1 is neither prime nor composite.

Analyzing the Given Numbers to Identify the Composite Number

We are given four numbers and need to identify which one is a composite number. Let's examine each option by looking for factors other than 1 and the number itself.

• Option 1: 51

  Let's check if 51 is divisible by any small prime numbers starting from 2.

  Is 51 divisible by 2? No, because it's an odd number.

  Is 51 divisible by 3? The sum of the digits of 51 is $5+1 = 6$. Since 6 is divisible by 3, 51 is also divisible by 3.

  We can perform the division: $51 \div 3 = 17$.

  So, the factors of 51 include 1, 3, 17, and 51. Since 51 has factors (3 and 17) other than 1 and 51, it is a composite number.

• Option 2: 41

  Let's check for factors of 41.

  Is 41 divisible by 2, 3, 5? No.

  We can check prime numbers up to the square root of 41. $\sqrt{41}$ is approximately 6.4. So we only need to check prime numbers less than or equal to 5 (i.e., 2, 3, 5).

  41 is not divisible by 2 (odd).

  41 is not divisible by 3 ($4+1=5$, not divisible by 3).

  41 is not divisible by 5 (doesn't end in 0 or 5).

  Since 41 is not divisible by any prime number less than or equal to its square root, 41 is a prime number. Its only factors are 1 and 41.

• Option 3: 61

  Let's check for factors of 61.

  The square root of 61 is approximately 7.8. We check prime numbers up to 7 (2, 3, 5, 7).

  61 is not divisible by 2 (odd).

  61 is not divisible by 3 ($6+1=7$, not divisible by 3).

  61 is not divisible by 5 (doesn't end in 0 or 5).

  61 is not divisible by 7 ($61 = 8 \times 7 + 5$).

  Since 61 is not divisible by any prime number less than or equal to its square root, 61 is a prime number. Its only factors are 1 and 61.

• Option 4: 31

  Let's check for factors of 31.

  The square root of 31 is approximately 5.5. We check prime numbers up to 5 (2, 3, 5).

  31 is not divisible by 2 (odd).

  31 is not divisible by 3 ($3+1=4$, not divisible by 3).

  31 is not divisible by 5 (doesn't end in 0 or 5).

  Since 31 is not divisible by any prime number less than or equal to its square root, 31 is a prime number. Its only factors are 1 and 31.

Based on our analysis, 51 is the only number among the options that has factors other than 1 and itself (specifically, 3 and 17). Therefore, 51 is a composite number.

Revision Table: Prime vs. Composite Numbers

Additional Information on Prime and Composite Numbers

Understanding prime and composite numbers is fundamental in number theory. Here are a few key points:

• Every integer greater than 1 is either a prime number or a composite number.
• The number 2 is the smallest and only even prime number.
• Composite numbers can be expressed as a product of prime factors. This is known as the Fundamental Theorem of Arithmetic. For 51, its prime factorization is $3 \times 17$.
• To check if a number N is prime, you only need to test for divisibility by prime numbers up to the square root of N. This significantly speeds up the process for larger numbers.