Number Theory and Fractions Review for Mastery Greatest Common Factor Holt

Did you know that all we accept to do to find the greatest common factor is piece of work the phrase GCF backwards?

Jenn, Founder Calcworkshop^®, 15+ Years Experience (Licensed & Certified Instructor)

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But kickoff, let'due south ascertain our terms.

Greatest Common Factor (GCF) — the largest number that divides evenly into each number in a given ready of numbers.

And at that place are two ways we can observe these factors, and they both involve using the letters One thousand-C-F?

Permit's smoothen a light on this with some examples…

Factor Pairs — In Action

And so, let'due south discover the GCF of 12 and 18 using our two techniques.

Our get-go method is to listing cistron pairs. We practise this by working G-C-F backward.

• F: List factors or factor pairs
• C: Place common factors
• 1000: Choose the greatest common factor

Cistron Pairs of 12 and eighteen

Notice that the largest number they both take in common is six — therefore, the GCF for these ii numbers is 6.

Only this technique is not foolproof, as it only works if you do place every factor for each number. The tried and true method is to use the prime number factorization method.

GCF Using Prime Factorization

In one case again, we will apply the letters from Thou-C-F to help us out. We will identify all the common factors, but instead of choosing the greatest, we will exist selecting the fewest mutual numbers.

Again, this is better seen in an instance.

So, permit's find the GCF of 12 and 18, but we will use the prime factorization method. This technique asks us to list all the prime number factors for each.

GCF of 12 and eighteen

The GCF is establish past identifying those numbers they have in mutual, and then selecting from those common numbers, which are the smallest — choosing the fewest they have in mutual as mentioned on Purple Math.

What is of import to annotation, is that both techniques (listing factors or prime number factorization) gave u.s. the same answer of 6 as the greatest common gene for 12 and eighteen. So, it really comes down to personal preference as to which method you want to use, and also the numbers that you are given to piece of work with.

What you will find is that the first method tends to be significantly faster than the second method. But as previously stated, the second method volition ever work.

Here'south a breakup of the two methods:

• When we list factors, we cull the GREATEST gene in common.
• When we apply prime factorization, we cull the FEWEST mutual factors.

Example — GCF of Three Numbers

Let'due south look at another example, finding the greatest common cistron of 3 numbers, showing both methods.

Method #one

Detect the GCF of 32, 48, and 64 past listing factors or factor pairs.

Greatest Common Factor of 32 — 36 — 64

Method #two

Find the GCF of 32, 48, and 64 by using prime factorizations.

Prime Factorization and GCF of 32 — 48 — 64

Find, once again, that both techniques gave u.s.a. a GCF of xvi.

What's important to note is that in Algebra, we will utilize both methods simultaneously when factoring polynomials. When factoring numerals (i.due east., numbers), we tend to use the first method of listing factor pairs. But when we gene variables, where we bargain with exponents, we utilize the prime factorization method.

So, knowing and being able to use both methods for finding the GCF is invaluable. And it's the hush-hush sauce for simplifying fractions too!

Worksheet (PDF) — Easily on Practise

Put that pencil to paper in these easy to follow worksheets — expand your noesis!

Greatest Common Factor — Practise Problems
Greatest Mutual Factor — Stride-by-Step Solutions

Video Tutorial — Full Lesson w/ Detailed Examples

In the video below, we will expand on these concepts, with additional examples increasing in complexity.

42 min

• Introduction to Video
• 00:00:26 – What is the GCF? How practise nosotros discover the GCF?

• 00:08:26 – Place the GCF using both methods (Examples #1-2)
• 00:15:33 – Make up one's mind the GCF of ii numbers (Examples #3-4)
• 00:22:23 – Find the greatest mutual factor of three numbers
• 00:28:sixteen – Use the listing cistron or prime factorization to solve the problem (Examples #seven-9)
• 00:39:00 – Discover the greatest mutual factor of three variables (Example #10)
• Practise Problems with Step-past-Step Solutions
• Affiliate Tests with Video Solutions

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Source: https://calcworkshop.com/fractions/greatest-common-factor/

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