Answer
Verified
Hint: To obtain the greatest common factor of two numbers we have to split each number in there factors which when multiplied again give the same number. We factorize both numbers and look for common factors in between them and the product of all common factors in the given number is known as the greatest common factor.
Complete step by step solution:
To find the greatest common factor of 12 and 15 we will firstly find the factor of both by using factorization
method.
For finding the factor of 12 we will use factorization method as below:
$\begin{align}
& 2\left| \!{\underline {\,
12 \,}} \right. \\
& 2\left| \!{\underline {\,
6 \,}} \right. \\
& 3\left| \!{\underline {\,
3 \,}} \right. \\
& \left| \!{\underline {\,
1 \,}} \right. \\
\end{align}$
So we get,
$12=2\times 2\times 3$
We can rewrite it as,
$12={{2}^{2}}\times
3$………$\left( 1 \right)$
In Similar way we will find the factor of 15 by using factorization method as below:
$\begin{align}
& 3\left| \!{\underline {\,
15 \,}} \right. \\
& 5\left| \!{\underline {\,
5 \,}} \right. \\
& \left| \!{\underline {\,
1 \,}} \right. \\
\end{align}$
So we get,
$15=3\times 5$………$\left( 2 \right)$
Now, if we compare equation (1) and (2) we get that there is only
one factor common in both which is 3.
Hence the greatest common factor of 12 and 15 is 3.
Note: The Factorization method is used when the greatest common factor is to be calculated for given numbers because GCF is always a prime number and only this method gives the prime factors of a number. If we are not getting a prime number as our GCF that means we have made some mistake in the calculation part. One thing that should be kept in mind while calculating the greatest common factor is that it can never be bigger than the given numbers.
The GCF of 12 and 15 is 3.
Steps to find GCF
- Find the prime factorization of 12
12 = 2 × 2 × 3 - Find the prime factorization of 15
15 = 3 × 5 - To find the GCF, multiply all the prime factors common to both numbers:
Therefore, GCF = 3
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GCF Calculator
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The greatest common factor (GCF) is also known as greatest common divisor (GCD) or highest common factor (HCF).
Are you on the hunt for the GCF of 12 and 15? Since you're on this page I'd guess so! In this quick guide, we'll walk you through how to calculate the greatest common factor for any numbers you need to check. Let's jump in!
Want to quickly learn or show students how to find the GCF of two or more numbers? Play this very quick and fun video now!
First off, if you're in a rush, here's the answer to the question "what is the GCF of 12 and 15?":
GCF of 12 and 15 = 3
What is the Greatest Common Factor?
Put simply, the GCF of a set of whole numbers is the largest positive integer (i.e whole number and not a decimal) that divides evenly into all of the numbers in the set. It's also commonly known as:
- Greatest Common Denominator (GCD)
- Highest Common Factor (HCF)
- Greatest Common Divisor (GCD)
There are a number of different ways to calculate the GCF of a set of numbers depending how many numbers you have and how large they are.
For smaller numbers you can simply look at the factors or multiples for each number and find the greatest common multiple of them.
For 12 and 15 those factors look like this:
- Factors for 12: 1, 2, 3, 4, 6, and 12
- Factors for 15: 1, 3, 5, and 15
As you can see when you list out the factors of each number, 3 is the greatest number that 12 and 15 divides into.
Prime Factors
As the numbers get larger, or you want to compare multiple numbers at the same time to find the GCF, you can see how listing out all of the factors would become too much. To fix this, you can use prime factors.
List out all of the prime factors for each number:
- Prime Factors for 12: 2, 2, and 3
- Prime Factors for 15: 3 and 5
Now that we have the list of prime factors, we need to find any which are common for each number.
In this case, there is only one common prime factor, 3. Since there are no others, the greatest common factor is this prime factor:
GCF = 3
Find the GCF Using Euclid's Algorithm
The final method for calculating the GCF of 12 and 15 is to use Euclid's algorithm. This is a more complicated way of calculating the greatest common factor and is really only used by GCD calculators.
If you want to learn more about the algorithm and perhaps try it yourself, take a look at the Wikipedia page.
Hopefully you've learned a little math today and understand how to calculate the GCD of numbers. Grab a pencil and paper and give it a try for yourself. (or just use our GCD calculator - we won't tell anyone!)
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