Question

What is the GCF of 15 and 28?

Answer 1

Hint: To find the greatest common factor of any two numbers or any $n$ numbers, we simply need to write the prime factorization of all the numbers one by one and then we need to pick the common factors out of all of them. And multiplying them will give us the greatest common factor of those numbers. Another way might be to list out all the factors of both the numbers and then find the greatest one out of them.

Complete step by step solution:
We are given two numbers 15 and 28. We can write the prime factorization of both of them. We do prime factorization by simply dividing the number continuously by the least prime number it is divisible by and we do this repeatedly until we get 1. We perform prime factorization on 15 first:
$15=3\times 5$
 Now, we write the prime factorization of 28:
$28=2\times 2\times 7$
We see that there are no common factors in prime factorization of both the numbers. When such a situation arises, we simply say that the greatest common factor of the numbers is 1. This happens because you can write the above statements as following too:
$15=3\times 5\times 1$
$28=2\times 2\times 7\times 1$
Now, 1 is common in both of these, so the greatest common factor of 15 and 28 is 1.

Note: It is a common mistake to multiply the factors obtained in prime factorization and then write that product as the greatest common factor, which is completely wrong. You are required to find the factor that is common but while you multiply like this you will be finding the least common multiple of them. Also, the following formula can be used to solve the problem:
$GCD\left(a,b\right)=\dfrac{a\times b}{LCM\left(a,b\right)}$
Here, the least common multiple of 15 and 28 is $15 \times 28$, hence
$GCD\left(15,28\right)=\dfrac{15\times 28}{15\times 28}=\dfrac{1}{1}=1$
If the numbers are too big then this formula might take a lot of time so better do prime factorization in that.