Problem 46 - Fundamentals of Mathematics

Question

Determine which of the whole numbers are prime and which are composite. 27

Step-by-Step Solution

Answer

27 is a composite number.

Step 1: Define Prime and Composite Numbers

A prime number is a whole number greater than 1 that has no positive divisors other than 1 and itself. A composite number is a whole number greater than 1 that has more than two positive divisors.

Step 2: Find Factors of 27

Determine the factors of 27 by checking the divisibility by numbers less than 27. The numbers 1, 3, 9, and 27 can divide evenly into 27.

Step 3: Count the Number of Factors

Since 27 has divisors 1, 3, 9, and 27, it has more than two divisors (1 and 27).

Step 4: Determine if 27 is Prime or Composite

Since the number 27 has more than two factors, it is a composite number.

Key Concepts

Factors of a Number

Factors of a number are the numbers you can multiply together to get the original number. They are like building blocks for numbers.
If you pick two numbers and multiply them, and they give you another number, those two numbers are factors of the big number.
Let's say you have the number 27, which was used in the exercise. You need to find out what numbers can be multiplied to become 27. What you do is you start checking numbers from 1 up.

• 1 is a factor of every number, so 1 is automatically a factor of 27.
• Next, you check 3. Since 3 times 9 equals 27, both 3 and 9 are factors.
• Then, you check further but find that only 27 times 1 equals 27, which brings us back to the beginning.

So, the factors of 27 are 1, 3, 9, and 27. Understanding factors is crucial since they are the keys to figuring out whether a number is prime or composite.

Prime Number Definition

A prime number is special in the world of numbers. Think of a prime number as a 'number trophy' that only few numbers can divide.
Prime numbers are unique because they have exactly two distinct positive factors: 1 and themselves. This means there is no other whole number that can evenly divide a prime number.
Here are some important points about prime numbers:

• They are always greater than 1.
• The number 2 is the smallest and the only even prime number because it is only divisible by 1 and 2.
• Prime numbers include 2, 3, 5, 7, 11, and so on.

If a number is prime, it cannot be split into smaller natural number factors, other than 1 and itself.

Composite Number Definition

Composite numbers are like big jigsaw puzzles that can be broken into smaller pieces.
These pieces are their factors. Put simply, a composite number has more than two distinct positive divisors.
When you find a number with more than just the two factors (1 and itself), that number is composite. For instance, consider 27 from the exercise:

• 27 can be divided evenly by 1, 3, 9, and 27.
• This gives it more than two factors.
• Thus, 27 is a composite number.

Composite numbers are always greater than 1 and include numbers like 4, 6, 8, 9, 10, 12, and so on. Recognizing composite numbers is easy once you find those extra divisors beyond just 1 and the number itself.