Problem 2 - Algebra A Combined Function

Question

List the factors of each number. See Examples 1 and \(2 .\) 8

Step-by-Step Solution

Answer

The factors of 8 are 1, 2, 4, and 8.

Step 1: Understand the Task

The task is to list all the factors of the number 8. Factors are numbers that divide the given number evenly (without leaving a remainder).

Step 2: Identify Possible Factors

Consider all integers starting from 1 up to 8 (since a factor of a number is always less than or equal to the number itself).

Step 3: Check for Divisibility

Check which of these integers divide 8 without a remainder: - **1** divides 8 with no remainder - **2** divides 8 with no remainder - **3** does not divide 8 evenly - **4** divides 8 with no remainder - **5** does not divide 8 evenly - **6** does not divide 8 evenly - **7** does not divide 8 evenly - **8** divides 8 with no remainder

Step 4: List the Factors

After checking the divisibility, the factors of 8 are 1, 2, 4, and 8.

Key Concepts

Prime Factors

Prime factors play a critical role in breaking down a number into its simplest building blocks. A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. When we factor a number into its prime factors, we express that number as a product of prime numbers. For instance, let's explore the prime factors of 8.

• Start by dividing 8 by the smallest prime number, which is 2. Since 8 is even, it's divisible by 2.
• We perform the division: 8 ÷ 2 = 4. The number 2 is a prime factor.
• Now take 4 and divide it again by 2 (since it's also even): 4 ÷ 2 = 2.
• Continue the division: 2 ÷ 2 = 1.

Once you reach 1, the process is complete. Every time 8 was divided by 2, 2 is a prime factor. Thus, the prime factorization of 8 is given by the expression: \[8 = 2 \times 2 \times 2 = 2^3\]

Divisibility

Divisibility is a fundamental concept when determining factors or solving various mathematical problems. A number is divisible by another if dividing them gives an integer quotient with no remainder.
Consider 8 and its divisibility by various numbers:

• 1: Any number is divisible by 1, including 8.
• 2: As 8 is an even number, it is divisible by 2, giving the quotient of 4.
• 3: Dividing 8 by 3 leaves a remainder, meaning 3 is not a factor.
• 4: Dividing 8 by 4 results in a clean division, providing a quotient of 2.
• 8: Since the number is dividing itself, the division results in 1, confirming that 8 is a factor of 8.

Understanding which numbers divide evenly helps in identifying all possible factors of a given number.

Integer Division

Integer division is the process of dividing two integers, where the result is rounded down to the nearest whole number. In other words, it's division without considering any remainder.
When finding factors, integer division helps us to check which numbers can divide another seamlessly. Let's revisit our example with the number 8:

• Dividing 8 by 1 using integer division yields 8, a whole number.
• 8 divided by 2 results in 4 - still an integer.
• Trying 8 divided by 3 gives a non-integer, demonstrating 3 is not a factor.
• Dividing 8 by 4 results in 2, confirming divisibility.
• The division of 8 by itself (8) yields 1.

Every integer that results from dividing a number using integer division without a remainder is a factor. Thus, understanding integer division is critical when listing all possible factors.