Vestrahorn: Stokksnes peninsula, Iceland
The Pitons: Soufriere, St. Lucia
Denali National Park and Preserve: Alaska
Ha Long Bay: Vietnam
Luberon Regional Nature Park: Provence, France
Monument Valley Navajo Tribal Park: Arizona and Utah
Great Barrier Reef: Australia
Mount Desert Island: Maine
Li River: Guilin, China
James Bond Island: Thailand
Milford Sound: Fiordland National Park, New Zealand
Plitvice Lakes National Park: Croatia
Greatest Common Factor (GCF)
The GCF of two or more non-zero integers, x, and y, is the
greatest positive integer m, which divides both x and y.
The greatest common factor is commonly known as GCF. Here,
greatest can be replaced with highest, and factor can be replaced
with divisor. So, the greatest common factor is also known as
Highest Common Divisor (HCD), Highest Common Factor (HCF),
or Greatest Common Divisor (GCD).
GCF is used almost all the time with fractions, which are used a
lot in everyday life. In order to simplify a fraction or a ratio, you
can find the GCF of the denominator and numerator and get the
required reduced form. Also, if we look around, the arrangement
of something into rows and columns, distribution and grouping, all
this require the understanding of GCF.
What is Greatest Common Factor (GCF)?
The GCF (Greatest Common Factor) of two or more numbers is
the greatest number among all the common factors of the given
numbers. The GCF of two natural numbers x and y is the largest
possible number that divides both x and y without leaving
any remainder. To calculate GCF, there are three common ways-
division, multiplication, and prime factorization.
Example: Let us find the greatest common factor of 18 and 27.
Solution:
First, we list the factors of 18 and 27 and then we find out the
common factors.
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 27: 1, 3, 9, 27
The common factors of 18 and 27 are 1, 3, and 9. Among these
numbers, 9 is the greatest (largest) number. Thus, the GCF of 18
and 27 is 9. This is written as: GCF(18, 27) = 9.
A factor of a number is its divisor as well. Hence the greatest
common factor is also called the Greatest Common Divisor (or)
GCD. In the above example, the greatest common divisor (GCD)
of 18 and 27 is 9 which can be written as:
GCD (18, 27) = 9.
How to Find GCF?
Following are the three methods for finding the greatest common
factor of two numbers:
Listing out common factors
Prime factorization
Division method
GCF by Listing Factors
In this method, factors of both the numbers can be listed, then it
becomes easy to check for the common factors. By marking the
common factors, we can choose the greatest one amongst all of
them. Let's look at the example given below:
Example: What is the GCF of 30 and 42?
Solution:
Step 1 - List out the factors of each number.
Step 2 - Mark all the common factors.
Step 3 - 6 is the common factor and the greatest one.
Therefore, GCF of 30 and 42 = 6. This method can be used for
finding GCF of three or more numbers as well.
Finding the greatest common factor by listing factors may be
difficult if the numbers are bigger. In such cases, we use
the prime factorization and division methods for finding GCF.
GCF by Prime Factorization
Prime factorization is a way of expressing a number as a product
of its prime factors, starting from the smallest prime factor of that
number. Let's look at the example given below:
Example: What is the GCF of 60 and 90?
Solution:
Step 1 - Represent the numbers in the prime factored form.
Step 2 - GCF is the product of the factors that are common to each of the given
numbers.
Thus, GCF (60,90) = 21 × 31× 51 = 30. Therefore, GCF of 60 and
90 = 30. We can also find the greatest common factor of three
numbers or more by this method.
Finding GCF by Division Method
The division is a method of grouping objects in equal groups,
whereas for large numbers we follow long division, which breaks
down a division problem into a series of easier steps. The
greatest common factor (GCF) of a set of whole numbers is the
largest positive integer that divides all the given numbers, without
leaving any remainder. Let's look at the example given below:
Example: Find the GCF of 198 and 360 using the division
method.
Solution:
Among the given two numbers, 360 is the larger number and 198
is the smaller number.
Step 1 - Divide the larger number by the smaller number using long division.
Step 2 - If the remainder is 0, then the divisor is the GCF. If the remainder is not 0,
then make the remainder of the above step as the divisor and the divisor of the above
step as the dividend and perform long division again.
Step 3 - If the remainder is 0, then the divisor of the last division is the GCF. If the
remainder is not 0, then we have to repeat step 2 until we get the remainder 0.
Therefore, the GCF of the given two numbers is the divisor of the
last division. In this case, the divisor of the last division is 18.
Therefore, the GCF of 198 and 360 is 18. This method is the most
appropriate method for finding GCF of large numbers. Let us see
how to use the division method to find the greatest common factor
of three numbers. In order to find the GCF of three numbers by
long division, the following steps are to be followed:
First, we will find the GCF of two of the numbers.
Next, we will find the GCF of the third number and the GCF of the first two numbers.
Example: Find the GCF of 126, 162, and 180.
First, we will find the GCF of the two numbers 126 and 162. [You
can choose any two numbers out of the given three numbers]
Thus, GCF of 126 and 162 = 18 ........(1).
Next, we will find the GCF of the third number, which is 180, and
the above GCF 18 by using the same method.
Thus, GCF of 180 and 18 = 18 ......(2).
From (1) and (2), GCF(126, 162, 180) = 18. Therefore, GCF of
126, 162, and 180 = 18.
GCF and LCM
The greatest common factor is the largest number that divides the
given numbers without leaving any remainder. On the other hand,
the LCM (least common multiple) Is the smallest
common multiple of the given numbers that can be divided by the
given numbers exactly, without leaving a remainder. For example,
let us find the GCF and LCM of numbers 6 and 8.
The factors of 6 are 1, 2, 3, 6, and the factors of 8 are 1, 2, 4, 8.
So, the common factors of 6 and 8 are 1 and 2, out of which 2 is
the highest common factor. Thus, GCF (6, 8) = 2. Now, the first
few multiples of 6 are 6, 12, 18, 24, 30, ..., and the first
few multiples of 8 are 8, 16, 24, 32, ... Out of these, the
least common multiple of 6 and 8 is 24. Thus, LCM (6, 8) = 24.
One very interesting relation between GCF and LCM of two
numbers is that the product of GCF and LCM of two numbers is
equal to the product of the numbers. For any two numbers a and
b, we have, LCM (a, b) × GCF (a, b) = a × b. Let us verify it using
the above example of 6 and 8. Let a = 6 and b = 8.
LCM (6, 8) × GCF (6, 8) = 6 × 8
24 × 2 = 6 × 8
48 = 48
Hence verified.
Now, let us learn the difference between GCF and LCM in the
section below.
Difference between GCF and LCM
The GCF or the greatest common factor of two or more numbers
is the greatest factor among all the common factors of the given
numbers, whereas the LCM or the least common multiple of two
or more numbers is the smallest number among all common
multiples of the given numbers. The following table shows the
difference between GCF and LCM:
Greatest Common Least Common
Factor(GCF) Multiple(LCM)
The GCF of two natural The LCM of two natural
numbers a and b is the greatest numbers a and b is the smallest
natural number x, which is a number y, which is a multiple of
factor of both a and b. both a and b.
In the intersection of the sets of In the intersection of the sets of
common factors, it is the common multiples, it is the
greatest value. minimum value.
Represented as, GCF(a,
b) = x Represented as, LCM(a, b) = y
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