Solution - Finding the greatest common factor with prime factorization

3

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Step-by-step explanation

1. Find the prime factors of 21

Tree view of the prime factors of 21: 3 and 7

The prime factors of 21 are 3 and 7.

2. Find the prime factors of 60

Tree view of the prime factors of 60: 2, 2, 3 and 5

The prime factors of 60 are 2, 2, 3 and 5.

3. Identify the common prime factors

Identify which of the prime factors all of the original numbers have in common:

Number	Prime factors
21	$3 \cdot 7$
60	$2 \cdot 2 \cdot 3 \cdot 5$

The common prime factor is 3

4. Calculate the GCF

The greatest common factor is equal to the product of the prime factors that all of the original numbers have in common.

GCF = $3$

The greatest common factor of 21 and 60 is 3.

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Why learn this

The common tasks of dividing, grouping, and distributing are applicable across an unlimited number of scenarios. Dividing a chocolate bar with ten squares among eight people; figuring out how much work each member of your project group should do; cutting squares out of a piece of cloth so there is none left over. These everyday actions all deal heavily with fractions, and to deal with fractions is to deal with greatest common factors (GCF).

The greatest common factor, which is sometimes referred to as the highest common factor (HCF) or the greatest common divisor (GCD), is the largest positive integer that a set of integers can all be divided by. Since fractions are commonly used in everyday life, and GCFs help us understand fractions, then, GCF can be helpful for understanding a wide variety of situations. For example, finding the GCF of a numerator and denominator can help us simplify very large fractions or ratios into smaller, more manageable numbers.

Terms and topics

Greatest common factor