Greatest Common Factor (GCF)
In short: The greatest common factor of several numbers is the biggest whole number that goes into each of them with nothing left over, like 12 for 48 and 84.
6th GradeNumbers and Operations6.NS.B.4
What is Greatest Common Factor?
The greatest common factor, or GCF, is the biggest number that two or more numbers share as a divisor. It tells you the largest equal groups you can make from several different totals at once.
Precisely, \(g\) is the greatest common factor of \(a\) and \(b\) when \(g\) divides both with no remainder and no larger whole number does. Six divisors are shared by \(48\) and \(84\), and the top one is 12.
How to spot it
- A problem asks for the largest equal groups, teams or packages made from two different totals.
- A fraction must be reduced to lowest terms by dividing top and bottom by the same number.
- You are asked to pull a common number out of a sum, as in \(48 + 84\).
- The words “greatest”, “largest” or “biggest” appear next to “common divisor” or “common factor”.
The method
- List all the factor pairs of each number, starting with 1 and working upward.
- Circle the factors that appear in every list.
- Pick the biggest circled number to name the greatest common factor; for 48 and 84 it is 12.
- Check by dividing: \(48 \div 12 = 4\) and \(84 \div 12 = 7\), and 4 and 7 share no factor besides 1.
Greatest Common Factor: solved example
Problem
A florist in Tucson has 72 roses and 90 tulips. She wants identical bouquets using every flower. What is the greatest number of bouquets she can make, and what does each hold?
Solution
The divisors of 72 run up to 36 and 72, and those of 90 run up to 45 and 90. Checking the big ones, 18 divides both: \(72 = 18 \cdot 4\) and \(90 = 18 \cdot 5\). Nothing larger works, since 4 and 5 share no factor.
She can make 18 bouquets, each with \(72 \div 18 = 4\) roses and \(90 \div 18 = 5\) tulips.
Try it
Question 1
Find the GCF of 30 and 45.
Show the answer
Both numbers split evenly by 15, and \(30 \div 15 = 2\) and \(45 \div 15 = 3\) share nothing more, so the greatest common factor is 15.
Question 2
Reduce \(\frac{56}{98}\) using the GCF.
Show the answer
The GCF of 56 and 98 is 14, so \(\frac{56 \div 14}{98 \div 14} = \frac{4}{7}\).
Greatest Common Factor vs. Least Common Multiple
This one is a divisor, so it is never larger than the smaller number: it splits the numbers down. The least common multiple is a multiple, so it is never smaller than the larger number: it builds them up. For 48 and 84, one is 12 and the other is 336.
Keep going
Lessons
Worksheets and quizzes
- Greatest Common Factor and Least Common Multiple – Worksheet
- Greatest Common Factor and Least Common Multiple – Quiz
Related words
Whole dictionary: A–Z · 6th Grade · Numbers and Operations