Problem 82 - Algebra 1: Concepts and Skills

Question

List all the factors of the number. 48

Step-by-Step Solution

Answer

The factors of the number 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. These numbers, when multiplied by an appropriate partner, yield a result of 48.

Step 1: Understand and Begin the Process

Factors are numbers that can evenly divide into another number. Let's start with the smallest factor, which is 1.

Step 2: Continue with Factor Finding

By dividing 48 by numbers from 1 to 48, when the division results in a whole number, that divisor is a factor. Continue until reaching number 48.

Step 3: List the Factors

Listing out all the calculated factors and check any missing values.

Key Concepts

Divisibility

Divisibility means that a number can be divided by another number without leaving a remainder. To determine if a number is divisible by another, we perform division and look at the result. If the result is a whole number, we say that divisibility is present. For example, we consider 48: it is divisible by 6 because 48 divided by 6 equals 8, which is a whole number. Understanding divisibility is crucial for finding all factors of a number.

Common rules of divisibility help us quickly determine if a number is a factor without performing long division. For instance, a number is divisible by 2 if its last digit is even, and by 5 if its last digit is 0 or 5. These rules simplify the process of finding factors and are especially useful when dealing with large numbers.

Integer Factors

Integer factors of a number are all the whole numbers that can divide that number without leaving a remainder. In other words, they perfectly 'fit into' the number.

When looking for the factors of 48, as in our original exercise, we start with 1 (since 1 is a factor of every number) and end with the number itself, as it divides itself perfectly. We proceed by testing each number between these two endpoints for divisibility. By systematically checking each integer, we eventually list the factors of 48, which include 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

Finding integer factors is essential for simplifying fractions, solving equations, and understanding number properties. Remember, every number has at least two factors: 1 and the number itself.

Prime Factorization

Prime factorization involves breaking down a number into its smallest building blocks—prime numbers, which are numbers greater than 1 that have no factors other than 1 and themselves. To perform prime factorization of a number like 48, we find the smallest prime number that divides 48 and continue the process with the quotient.

Let's break it into steps for 48: firstly, 48 is even, so it's divisible by 2. Dividing 48 by 2, we get 24. Continuing this process, we find that 24 is also divisible by 2, giving us 12. This process is repeated until we can no longer divide by prime numbers. Ultimately, 48 can be expressed as 2 x 2 x 2 x 2 x 3, or simply as \(2^4\times3\).

Prime factorization is vital for various mathematical applications, including finding the greatest common factor, least common multiple, and simplifying radical expressions.