Problem 54 - Algebra 1: Concepts and Skills

Question

List all the common factors of the pair of numbers. $$ 4,28 $$

Step-by-Step Solution

Answer

The common factors of 4 and 28 are 1, 2 and 4.

Step 1: Find the factors of each number

Start with the first number, which is 4. The factors of 4 are 1, 2 and 4. Do the same with the second number, 28. The factors of 28 are 1, 2, 4, 7, 14 and 28.

Step 2: Identify common factors

Look at the factors lists for both numbers and find the numbers that are present on both lists. These are the common factors. In this case, the common factors are 1, 2 and 4.

Key Concepts

Factors of a Number

Factors of a number are the integers that you can multiply together to produce that number. To find the factors of any number, you need to look for all the whole numbers that divide the original number without leaving a remainder.
Let's take the number 4 as an example. You can think of it as:\[4 = 1 \times 4\] and \[4 = 2 \times 2\].
Thus, the factors of 4 are 1, 2, and 4. Similarly, for the number 28, you can find that:\[28 = 1 \times 28\], \[28 = 2 \times 14\], and \[28 = 4 \times 7\].
So, the factors of 28 are 1, 2, 4, 7, 14, and 28.
To determine the factors of a number, follow these steps:

• Start with the number 1 and the number itself.
• Check all numbers between 1 and the original number to see if they divide evenly.
• Include divisors without a remainder in your list.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) of two or more numbers is the largest number that divides all the given numbers without any remainder. This is also known as the Greatest Common Divisor (GCD).
To find the GCF, you first list the factors of each number and then identify the largest factor that is common to each list.
For example, let’s find the GCF of 4 and 28:

• Factors of 4 are: 1, 2, 4
• Factors of 28 are: 1, 2, 4, 7, 14, 28

The common factors of 4 and 28 are 1, 2, and 4. From these, the largest number is 4. So, the GCF of 4 and 28 is 4.
This concept is useful for simplifying fractions or solving problems involving integers. Calculating the GCF is an essential skill in number theory and lays the groundwork for understanding more complicated mathematical concepts.

Number Theory

Number theory is a branch of pure mathematics devoted to the study of the integers and more generally integer-valued functions. It revolves around the properties and relationships of numbers, particularly factors, multiples, and prime numbers.
In number theory, understanding the concept of factors and the greatest common factor (GCF) is crucial. These concepts play a fundamental role in solving problems related to divisibility, simplification, and the structural formation of numbers.
Number theory has practical applications too. It aids in cryptography, computer science, and coding theory amongst others. Here are some simple number theory concepts that relate to factors:

• Prime Numbers: These are numbers that have only two factors: 1 and themselves.
• Composite Numbers: Numbers that have more than two factors.
• Divisibility Rules: Simple rules to easily determine if one number is divisible by another.

By applying the principles of number theory, you can develop problem-solving strategies and improve your mathematical reasoning skills. Exploring the richness of numbers goes beyond elementary arithmetic and opens the door to deeper mathematical discoveries.