Problem 57 - Algebra 1: Concepts and Skills

Question

List all the common factors of the pair of numbers. $$ 12,30 $$

Step-by-Step Solution

Answer

The common factors of 12 and 30 are 1, 2, 3, and 6.

Step 1: Find Factors of 12

The factors of 12 are: 1, 2, 3, 4, 6, and 12. These are the numbers that can divide 12 without leaving a remainder.

Step 2: Find Factors of 30

The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, and 30. These are the numbers that can divide 30 without leaving a remainder.

Step 3: Identifying Common Factors

By examining the list of factors for both 12 and 30, it is clear that the common factors, i.e., numbers that appear on both lists, are: 1, 2, 3, and 6.

Key Concepts

Factors of Numbers

When we talk about the factors of a number, we're discussing the numbers that divide it exactly, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. This means each of these numbers can divide 12 evenly. To find factors, we generally start by testing the smallest numbers, beginning with 1 and working our way up. For this, we check every integer up to half of the number because no integer larger than half of a number, except the number itself, can be a factor. A quick reminder:

• Factors are whole numbers
• They always include 1 and the number itself
• They must divide the original number evenly

There's a certain elegance in factors as they reveal numerical relationships inherent in numbers themselves.

Greatest Common Divisor

The Greatest Common Divisor, often abbreviated as GCD, is the largest number that can perfectly divide a set of numbers. In the context of the numbers 12 and 30, we've determined their common factors to be 1, 2, 3, and 6. Among these, the greatest is 6, which means the GCD of 12 and 30 is 6. Understanding the GCD is vital in simplification processes, helping with fraction reduction and problem-solving in algebra. To determine the GCD:

• List the factors of each number
• Identify the common factors
• Select the largest common factor

The GCD simplifies complex mathematical expressions and ensures clearer calculations, broadening our numerical comprehension.

Divisibility Rules

Divisibility rules are tools that help quickly determine if one number can be divided by another without leaving a remainder. These rules simplify the process of finding factors without needing to perform division. Here are some basic divisibility rules:

• A number is divisible by 2 if it is even.
• It is divisible by 3 if the sum of its digits is a multiple of 3.
• Divisibility by 5 happens if the number ends in 0 or 5.

Applying these rules to determine factors can make finding common factors and GCDs much smoother. By using divisibility rules, even complex mathematical challenges can be tackled more straightforwardly, fostering ease and speed in calculations.