Problem 9 - Fundamentals of Mathematics

Question

Determine which of the following whole numbers are prime and which are composite. 16

Step-by-Step Solution

Answer

16 is a composite number.

Step 1: Define Prime and Composite Numbers

A prime number is a whole number greater than 1 that has no divisors other than 1 and itself. A composite number, on the other hand, has divisors other than 1 and itself, meaning it can be divided evenly by numbers other than 1 and itself.

Step 2: Evaluate the Given Number

Look at the number 16. To determine if it is prime or composite, check if it has divisors other than 1 and itself.

Step 3: Check Divisibility

Evaluate the divisibility of 16. Aside from itself (16) and 1, check if there are other numbers that divide 16 evenly, i.e., with no remainder.

Step 4: List Divisors

Determine the list of divisors for 16: 1, 2, 4, 8, and 16. Notice that there are numbers other than 1 and 16 that divide 16 evenly.

Step 5: Conclusion

Since 16 has divisors other than 1 and itself, it is identified as a composite number.

Key Concepts

Divisibility

Divisibility is a fundamental concept in mathematics that determines if one number can be divided by another without leaving a remainder. It helps us understand how numbers can be broken down into factors. When we talk about divisibility, it means checking if a smaller number can perfectly fit into a larger number without leaving any leftover parts. For example, the number 16 can be divided by 2 because it fits perfectly 8 times, leaving no remainder. Similarly, it can also be divided by 4, because 4 fits exactly 4 times into 16.

• To say a number is divisible by another, it means perfect division with no remains.
• Divisibility helps us figure out if a number has more divisors apart from 1 and itself, which is crucial in identifying composite numbers.

Grasping divisibility rules helps students simplify numbers and find factors efficiently, making it essential for understanding the difference between prime and composite numbers.

Whole Numbers

Whole numbers are the set of numbers that include zero and all positive numbers without fractions or decimals. They are numbers that you can count on your fingers and toes. These include numbers like 0, 1, 2, 3, and so on. Whole numbers are essential in basic arithmetic operations and everyday calculations. They set a foundation for exploring other number sets.

• Whole numbers begin at zero and continue indefinitely.
• They do not include negative values or fractions.

Understanding whole numbers is vital as prime and composite numbers fall within this category. When discussing factors and divisors, we primarily refer to whole numbers.

Factors

Factors are numbers that evenly divide another number, leaving no remainder. They are like the building blocks of whole numbers. For example, in the case of 16, its factors are 1, 2, 4, 8, and 16. Each of these numbers can divide 16 perfectly without leaving any remainders.

• Finding factors involves dividing and checking divisibility, ensuring there's no remainder.
• Every number has at least two factors: 1 and itself.

Factors are key to deciding if a number is prime or composite. Once we know all the factors of a number, we can quickly see which numbers divide the number perfectly, helping to classify the number accordingly.

Prime Number Definition

Prime numbers are special whole numbers greater than 1 that have exactly two distinct positive divisors: 1 and the number itself. This means a prime number cannot be divided by any other whole number without leaving a remainder. Understanding this definition is crucial in distinguishing prime numbers from composite ones.

• Examples of prime numbers include 2, 3, 5, 7, and 11.
• The number 2 is the only even prime number because any even number greater than 2 can be divided by 2.

Recognizing prime numbers involves checking if the number has only two factors. Understanding their unique definition helps students analyze numbers effectively for divisibility, determining if they are prime or composite.