Problem 53 - Algebra 1: Concepts and Skills

Question

List all the common factors of the pair of numbers. $$ 3,21 $$

Step-by-Step Solution

Answer

Common factors of 3 and 21 are: 1 and 3.

Step 1: List down the factors

First, we will list down all the factors of both 3 and 21. Factors of 3 are: 1 and 3. Factors of 21 are: 1, 3, 7 and 21.

Step 2: Identify common factors

Next, find the common factors from both lists. The common factors of 3 and 21 are: 1 and 3.

Key Concepts

Factors

Factors are numbers that divide another number evenly, without leaving a remainder. For any given number, factors are the integers that you can multiply together to reach that number. Let's consider the number 6. The factors of 6 are 1, 2, 3, and 6, since:

• 1 × 6 = 6
• 2 × 3 = 6

Finding factors of a number involves trying to divide it by smaller integers. If it divides equally, then the divisor is a factor. Another point to remember is that factors come in pairs. This means that for every factor below a number, there is a corresponding factor that is the result of dividing the number by the smaller factor.

Pairs of Numbers

The concept of pairs of numbers plays a significant role when discussing factors because each factor can relate to another to form a multiplying pair. For example, when we list the factors of a number like 12, we find pairs such as:

• 1 and 12 (1 × 12 = 12)
• 2 and 6 (2 × 6 = 12)
• 3 and 4 (3 × 4 = 12)

These pairs demonstrate how numbers work together to multiply to the original number. Understanding pairs is helpful not just in factorization but also in solving problems involving multiplying two numbers. When finding common factors between pairs of numbers, it essentially means identifying shared elements from each number’s factor list.

Step-by-Step Solution

The step-by-step process is crucial for breaking down problems into smaller, manageable tasks. In this exercise, the goal is to find common factors, and here's how you can do that efficiently:
1. **List the Factors**: Start by identifying factors for each number. As shown, for 3, the factors are 1 and 3. For 21, the factors are 1, 3, 7, and 21.
2. **Find Commonalities**: Compare the lists to find common factors. Here, the shared factors are 1 and 3.
3. **Verification**: Check if these common factors divide both numbers without a remainder. For both 3 and 21, 1 and 3 indeed divide them completely.

• This method works universally for any pair of numbers and is a fundamental skill in arithmetic.
• It also lays the foundation for more complex mathematical concepts, like finding the greatest common divisor (GCD).