Problem 53 - Fundamentals of Mathematics

Question

Determine which of the whole numbers are prime and which are composite. 34

Step-by-Step Solution

Answer

34 is a composite number.

Step 1: Define Prime and Composite Numbers

A prime number is a whole number greater than 1 that is only divisible by 1 and itself. A composite number is a whole number greater than 1 that has more divisors than just 1 and itself.

Step 2: Check Divisibility by 2

Since 34 is an even number, it is divisible by 2. Therefore, 34 can be divided by 2 to give a quotient of 17.

Step 3: Conclude Based on Divisibility

Since 34 is divisible by a number other than 1 and itself (it is divisible by 2), 34 is not a prime number. It is a composite number.

Key Concepts

Divisibility

To determine whether a number is prime or composite, we need to check its divisibility. Divisibility refers to whether a number can be divided by another number without leaving a remainder. If a number can be divided exactly by another, it means it is divisible by that number.

For instance, consider the number 34. If we say 34 is divisible by 2, it means when you divide 34 by 2, you get a whole number without a remainder, which in this case is 17. This indicates that \[34 \div 2 = 17\].

Divisibility is crucial in determining whether a number is prime or composite because:

• If a number has only two divisors, 1 and itself, it is a prime number.
• If a number has more than two divisors, it is a composite number.

For example, 34 is also divisible by 1 and 17, confirming it as a composite number.

Prime Number Definition

A prime number is a special number in mathematics. It is defined as a whole number greater than 1 that has no divisors other than 1 and itself. This means that no other whole number can divide them evenly without leaving a remainder.

Here are some characteristics of prime numbers:

• They are greater than 1.
• They have exactly two distinct divisors: 1 and the number itself.

For example, consider the number 3:

• It can be divided evenly by 1 and 3.
• There are no other numbers that divide 3 evenly.

Thus, 3 is a prime number. Keep this rule in mind when checking if a number is prime.

Composite Number Definition

Composite numbers are the opposite of prime numbers. They are whole numbers greater than 1 that have more than two distinct divisors. In simple terms, if a number can be divided evenly by numbers other than 1 and itself, it is composite.

Some features of composite numbers include:

• They are greater than 1.
• They have more than two divisors.

For example, let's revisit the number 34:

• 34 is divisible by 1, 2, 17, and itself (34).
• This means it has four divisors, which makes it a composite number.

Identifying composite numbers can be useful, as they can be broken down into smaller factors, unlike prime numbers.