Problem 14 - Fundamentals of Mathematics

Question

Find the greatest common factor (GCF) of the numbers. 66 and 165

Step-by-Step Solution

Answer

The GCF of 66 and 165 is 33.

Step 1: Define GCF

The greatest common factor, or GCF, of two numbers is the largest number that divides both of them without leaving a remainder.

Step 2: List Factors of 66

To find the factors of 66, list all integers that divide 66 without leaving a remainder. These are: 1, 2, 3, 6, 11, 22, 33, and 66.

Step 3: List Factors of 165

Similarly, list all the factors of 165. These are: 1, 3, 5, 11, 15, 33, 55, and 165.

Step 4: Identify Common Factors

Identify the common factors from the lists obtained in Steps 2 and 3. The common factors of 66 and 165 are: 1, 3, 11, and 33.

Step 5: Select Greatest Common Factor

Among the common factors identified, select the greatest one. Thus, the greatest common factor of 66 and 165 is 33.

Key Concepts

Divisibility

Divisibility is an important concept in mathematics. It's all about determining whether one number can be divided by another without leaving a remainder. For instance, when we say a number is divisible by another, we mean that dividing them results in a whole number.

• For number 66, its divisibility is tested by checking each potential divisor like 1, 2, 3, and so on, all the way up to itself.
• Similarly, 165 is tested against various numbers, like 1, 3, 5, and others, to see which ones divide it perfectly.

Understanding divisibility helps you see the pattern of numbers and the relationships between them, which is crucial in finding the greatest common factors.

Integer Factorization

Integer factorization is a process seeing which numbers multiply together to equal the original number. It breaks down a number into its basic building blocks. Think of it like taking apart a Lego structure to see exactly which pieces were used.

• In the case of 66, you find the numbers that, when multiplied, will form 66. That's why you list its factors as 1, 2, 3, 6, 11, 22, 33, 66.
• Likewise, for 165, the goal is to see what numbers multiply to become 165, leading to factors like 1, 3, 5, 11, 15, 33, 55, 165.

Once you have all the factors, it becomes much easier to spot commonalities between different numbers, a step necessary for solving problems involving greatest common factors.

Mathematics Problem Solving

Mathematics problem solving skills are like a toolset for tackling various numerical problems. Here, with the example of finding the greatest common factor, these skills involve logical thinking and strategy.

• The process starts by understanding the problem, which, in this instance, is to identify the largest number that divides both given numbers (66 and 165) without a remainder.
• Once that's clear, you systematically list the factors of each number and then identify the common factors.
• Finally, choosing the largest from the list of common factors, in this case 33, provides the solution.

This structured approach ensures that every step is considered, leading to an accurate solution, and develops a valuable problem-solving mindset that is applicable beyond just mathematics.