Finding Greatest Common Factor by Prime Factorization

We received a question regarding the 2nd method of finding Greatest Common Factor by Prime Factorization in the Math In Focus Grade 4 materials. This example is from Math In Focus workbook 4A page 23,

The “Method 2” mentioned above uses prime factorization as a procedure to find the GCF between two numbers.

Procedure

• In each step, divide the numbers by a simple prime factor common to both, e.g. 2, 3.
• Repeat until the resulting numbers do not have common factors.
• Multiply all common prime factors to obtain the greatest common factor between the two numbers.

For example, for the numbers 32 and 56,

Step 1: Divide both 32 and 56 by 2, resulting quotients are 16 and 28
Step 2: Divide both 16 and 28 by 2, resulting quotients are 8 and 14
Step 3: Divide both 8 and 14 by 2, resulting quotients are 4 and 7
Since 4 and 7 do not have any further common factors, first part of the procedure stops here.

To find the greatest common factor, multiply the 3 common prime factors

2 x 2 x 2 = 8

Why It Works

The method finds the common prime factors between two numbers sequentially, resulting in prime factorizations

32 = 2³x2²

56 = 2³x7¹

Hence by inspection, the greatest of them is the product of all three, i.e.

2 x 2 x 2 = 8

The method will also work if we use non-prime numbers for any particular step, e.g. 4 and 2 in 2 steps, or direct 8 in one step, if the student can see it mentally, because the goal is to find the greatest common factor and not each individual prime factors.

When are Factors and GCF introduced?

Common Core

In the US Common Core standards, factors are explicitly mentioned at two levels. The 4th grade common core 4.OA standard 4 requires students to be able to:

Gain familiarity with factors and multiples.
4. Find all factor pairs for a whole number in the range 1–100. Recognize
that a whole number is a multiple of each of its factors. Determine
whether a given whole number in the range 1–100 is a multiple of a
given one-digit number. Determine whether a given whole number in
the range 1–100 is prime or composite.

The 6th grade common core 6.NS standard 4 requires students to be able to:

Compute fluently with multi-digit numbers and find common factors and multiples.
4. Find the greatest common factor of two whole numbers less than or
equal to 100 and the least common multiple of two whole numbers
less than or equal to 12. Use the distributive property to express a
sum of two whole numbers 1–100 with a common factor as a multiple
of a sum of two whole numbers with no common factor. For example,
express 36 + 8 as 4 (9 + 2).

Math In Focus

As we saw in the example above, the US edition of Math In Focus introduces two methods of finding the GCF in grade 4. The chapter itself mainly introduced the concept of factors and the two methods of finding GCF are presented as a boxed side example, in between exercises to get students familiar with factors, prime and composite numbers.

Singapore Education System

In Singapore, the term product is introduced in 3rd grade, factors and multiples are introduced in 4th grade. Greatest Common Factor (GCF) and Least Common Multiple (LCM) in only formally introduced in 7th grade (from Teaching to Mastery Mathematics: Teaching of Fractions).

Our Views

In our view, procedural methods for finding GCF should not be mandatory for all 4th grade level instruction, and the topic of GCF in general should be introduced only after students fully understood factors. While working on finding factors of a number, it is very natural to ask if two numbers have factors in common and then extending to ask which of the common factors is the greatest. Hence, the topic of GCF can be mentioned in the context of factors, and students are generally excited to know that they are learning a concept that will be taught in 6th grade.

Coming Up

Next week, we will write about why learning GCF is important, its application in more advanced topics and what we educators should take note of when teaching the topic.

Thank you!

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