Problem 20 - Elementary and Intermediate Algebra

Question

Label each of the following numbers as prime, composite, or neither. $$ 75 $$

Step-by-Step Solution

Answer

Composite

Step 1: Understand the Terms

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 natural number greater than 1 that has more than two positive divisors. A number that is neither prime nor composite is usually 0 or 1.

Step 2: Check the Number 75

First, note that 75 is greater than 1, so it cannot be neither. To determine if it is prime or composite, we need to check its divisors.

Step 3: Divisibility Test

Check for divisors of 75. If it has divisors other than 1 and itself, then it is composite.

Step 4: Identify Divisors

75 is divisible by 3 (since the sum of its digits, 7 + 5 = 12, is divisible by 3). It is also obviously divisible by 5 since it ends in a 5. Therefore, 75 has divisors other than 1 and itself.

Step 5: Conclusion

Since 75 has divisors other than 1 and itself (1, 3, 5, 15, 25, and 75), it is categorized as a composite number.

Key Concepts

prime numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means a prime number cannot be divided evenly by any other number except for 1 and its own value. For example, the number 7 is a prime number because the only numbers that divide 7 evenly are 1 and 7 itself. No other numbers can divide 7 without leaving a remainder. Prime numbers are like building blocks in mathematics because every natural number greater than 1 can be uniquely factored into prime numbers. Here are a few key characteristics of prime numbers:

• They are greater than 1.
• The only positive divisors they have are 1 and the number itself.
• Examples of prime numbers include 2, 3, 5, 7, 11, and 13.

The number 2 is also unique as it is the only even prime number. All other even numbers can be divided by 2, and hence, cannot be prime.

composite numbers

Composite numbers are natural numbers greater than 1 that have more than two positive divisors. This means that in addition to being divisible by 1 and the number itself, they can also be divided by at least one more number. Take the number 75 from our example:

• It can be divided by 1, 3, 5, 15, 25, and 75.
• This indicates that 75 has multiple divisors besides 1 and itself, qualifying it as a composite number.

Composite numbers can be thought of as 'non-prime' numbers because they have factors other than just 1 and the number itself. Notice that:

• All even numbers greater than 2 are composite since they can be divided by 2.
• Examples of composite numbers include 4, 6, 8, 9, 10, and 12.

Essentially, composite numbers are formed by multiplying prime numbers together.

divisibility rules

Divisibility rules help us quickly determine whether one number can be divided by another without performing full division. These rules make it easy to identify factors of a number, which helps categorize them as either prime or composite. For instance:

• A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 5 if it ends in 0 or 5.

Using these rules, we quickly saw that 75 is composite:

• The sum of its digits, 7 + 5, equals 12, which is divisible by 3.
• Therefore, 75 is divisible by 3.

Furthermore, since 75 ends in a 5, it conforms to the rule for divisibility by 5, confirming more divisors. Divisibility rules simplify the process of identifying whether numbers are prime or composite without tedious calculations.

natural numbers

Natural numbers are the set of positive integers starting from 1 and counting upwards. This set can be represented mathematically as: \[ \text{Natural Numbers} = \{1, 2, 3, 4, 5, \ldots\} \]. Natural numbers are significant in counting and ordering. When we discuss prime and composite numbers, we're only considering natural numbers greater than 1:

• Prime numbers are those natural numbers greater than 1 which have no divisors other than 1 and themselves.
• Composite numbers are those natural numbers greater than 1 which possess more than two divisors.

It is important to note that the natural numbers include 1, but 1 is neither prime nor composite. Instead, it serves as the fundamental unit of counting but doesn't meet the criteria of having exactly two distinct positive divisors, which is required for prime numbers. Similarly, since it doesn't have more than two divisors, it isn't composite either.