Problem 70 - Algebra 1: Concepts and Skills

Question

Determine whether the number is prime or composite. If it is composite, list all of its factors. (Skills Review p. 761) $$15$$

Step-by-Step Solution

Answer

The number 15 is a composite number. Its factors are 1, 3, 5, and 15.

Step 1: Prime or Composite

Start by testing whether the number 15 is prime. A prime number has only two distinct natural divisors, which are 1 and the number itself. Therefore, if the number can be divided by any other numbers apart from 1 and itself, it is not a prime number, it is composite.

Step 2: Test Divisibility

Divide 15 by all numbers up to 15 to check if it has more divisors apart from 1 and itself. We find that 15 is divisible by 3 and 5, other than 1 and 15. Hence, 15 is not a prime number, it's a composite number.

Step 3: List all Factors

Now, we list down all the integers that can exactly divide the number 15. These numbers are its factors. For 15, the factors are 1, 3, 5, and 15.

Key Concepts

Divisibility Rules

Divisibility rules are shortcuts that help us determine whether a number can be divided by another without performing long division. For example, a number is divisible by 2 if its last digit is even, and it's divisible by 3 if the sum of its digits is a multiple of 3. Knowing these rules is handy for quickly spotting the factors of a number.

Let's look at the number 15: It ends with 5, so according to the divisibility rules, it's divisible by 5. Furthermore, the sum of its digits (1 + 5) equals 6, which is a multiple of 3, indicating that 15 is also divisible by 3. This tells us immediately, without any lengthy calculations, that 15 is not a prime number.

Prime Number Definition

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, it cannot be formed by multiplying two smaller natural numbers. For instance, numbers like 2, 3, 5, and 7 are prime because there are no other numbers that can multiply together to give these numbers, except for 1 and the number itself.

In the case of the number 15, we determine that it is not a prime number because it has divisors other than 1 and itself. Specifically, 15 is divisible by 3 and 5, making it a composite number with multiple factors.

Listing Factors

To list the factors of a number, we need to find all whole numbers that divide that number exactly without leaving a remainder. In our example, we want to list all factors of 15. One efficient method is to start with the number 1 and the given number itself (which are always factors), and then check for divisibility with numbers between them.

Starting from 1, moving up, we hit 3, which divides 15 to give us 5. Both 3 and 5 are factors of 15. We continue until we reach the half of 15, knowing that there cannot be a factor greater than half the number except the number itself. Hence, the list of factors for the number 15 is complete with 1, 3, 5, and 15.

Fundamental Concept of Arithmetic

The fundamental concept of arithmetic, also known as the fundamental theorem of arithmetic, states that every integer greater than 1 can be expressed as a product of prime numbers in a manner that is unique, up to the order of the factors. This is the foundation of number theory and has significant implications in various areas of mathematics.

When applying this concept to the example of 15, we express it as a product of its prime factors: 15 can be written as 3 times 5. Both 3 and 5 are prime, which means they cannot be broken down further into smaller factors. This expression of 15 demonstrates the unique prime factorization that is at the heart of the fundamental concept of arithmetic.