Question

What is the greatest common factor (GCF) of $321$ and $146$?

Answer 1

Hint: There are various methods for finding the greatest common divisor of the given numbers. The simplest method to find the greatest common divisor is by prime factorization method. In the prime factorization method, we first represent the given two numbers as a product of their prime factors and then find the product of the lowest powers of all the common factors.

Complete step by step solution:
In the given question, we are required to find the highest common factor of $321$ and $146$.
To find the highest common factor of the given numbers: $321$ and $146$, first we find out the prime factors of all the numbers.
We know that both the numbers given to us, $321$ and $146$, are composite numbers. Composite numbers are numbers that are divisible by a number other than one and the number itself. They have more than two factors.
So, we do the prime factorization of the numbers as,
Prime factors of $321$$ = 3 \times 107$
Prime factors of $146$$ = 2 \times 73$
Now, the greatest common divisor is the product of the lowest powers of all the common factors.
Now, we can see that there is no repeated factor.
Hence, the greatest common factor of the numbers $321$ and $146$ is $1$.

Note: Highest common factor or the greatest common divisor is the greatest number that divides both the given numbers. Similarly, the highest common factor can also be found by using prime factorization methods as well as using Euclid’s division lemma. Highest common divisor is just a product of common factors with the lowest power.
Using the Euclid’s Division lemma, we try to find the combination of unique numbers q and r such that $a = bq + r$, where $0 \leqslant r < b$.
Here, $a = 321$ and $b = 146$.
So, we get,
$321 = 146 \times 2 + 29$
Now, the r in the above step becomes the q for the next step and b in the previous step becomes the a for the next step.
Hence, we get,
$146 = 29 \times 5 + 1$
Now, the r in the above step becomes the q for the next step and b in the previous step becomes the a for the next step.
$29 = 1 \times 29 + 0$
Now, as we observe that the remainder r is zero. So, we can conclude that the number b in the last step is the greatest common factor. Hence, we get the GCF of $321$ and $146$ as $1$.