Problem 76 - 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) $$64$$

Step-by-Step Solution

Answer

The number 64 is a composite number. The factors of 64 are: 1, 2, 4, 8, 16, 32, 64.

Step 1: Identify if the number is Prime or Composite

A number is prime if it has exactly two distinct divisors: 1 and itself. However, if the number has more than two distinct divisors, it's a composite number. In case of 64, you can see that it can be divided by numbers other than 1 and 64. Therefore, 64 is a Composite number.

Step 2: Determine the factors of the Number

Factors of a number are the numbers that divide the number exactly. Start by dividing 64 by 2, if the remainder is 0 then 2 is a factor of 64. Keep doing this for all the numbers up to 64. The factors of 64 are: 1, 2, 4, 8, 16, 32, 64.

Key Concepts

Understanding Factors

Factors are the numbers that exactly divide a given number. They're essential building blocks in understanding numbers. In simple terms, if you take a number and see that it can be divided evenly by another number, that other number is a factor. For example, the number 64 can be divided evenly by several numbers, meaning it has multiple factors.

• Each number is a factor of itself and 1 is a factor of every number.
• Factors come in pairs, because they multiply to form the number. For 64, when you multiply 1 and 64, or 2 and 32, you get 64.
• Finding factors requires identifying all numbers that divide evenly into the original number.

To determine the factors of 64, you start by taking the smallest numbers, like 2 and then check larger numbers like 8 or 16 to see if they divide 64 evenly, without any remainder.

Divisors and their Role

Divisors are almost like aliases for factors. A divisor of a number is any number that divides it without leaving a remainder. Essentially, being a divisor means it "goes into" the number perfectly, without any leftover.

• All factors are divisors, but we also use the term divisors to imply the action of division.
• A number always has at least two divisors: 1 and itself. These divisors are shared by all numbers.
• For example, the divisors of 64 include 1, 2, 4, 8, 16, 32, and 64. All these numbers divide 64 exactly.

By listing these divisors, you simplify tasks like finding the greatest common divisor (GCD) or simplifying fractions.

Composite Numbers Explained

Composite numbers are numbers that have more than just two divisors. Unlike prime numbers, which only have 1 and themselves as divisors, composites have additional factors.

• A composite number can be thought of as having a richer set of divisors.
• In the number 64, we see it has many divisors: 1, 2, 4, 8, 16, 32, and 64. Thus, it's classified as composite.
• Any even number greater than 2 is composite because it can be divided by at least 1, 2, and itself.

Recognizing composite numbers is practical when factoring large numbers or simplifying mathematical expressions. Knowing that 64 is composite helps with operations such as breaking down numbers or identifying patterns within mathematics.