Question

What are the greatest common factors of $420$ and \[660\].

Answer 1

Hint: We should know the prime factor of each number to find the greatest common factor. It is possible to factor the figures into different combinations. If you are familiar with the multiplication tables, you can easily figure out the factors. A number that has only two factors and the number itself. Each given number is written as the product of prime numbers and then the product of each common prime factor with the smallest power is found.

Formula Used:
Greatest Common Factor $(a,b)$$ \times $Least Common factor $(a,b)$$ = a,b$
Where $a$ and $b$ are positive integer,
Least common multiple,
\[LCM = \dfrac{{a \times b}}{{\gcd (a,b)}}\]
Greatest common multiple or Highest common factor
$GCD = \dfrac{{a \times b}}{a}\,$and $\dfrac{{a \times b}}{b}$
Where $a$ and $b$ are positive integer,
Two numbers are the largest number that divides both the two numbers exactly.

Complete step-by-step answer:
To find the prime factors of \[420\]
We get,
$420 = 2 \times 2 \times 3 \times 5 \times 7$
Find the prime factor of $660$
We get,
$660 = 2 \times 2 \times 3 \times 5 \times 11$
We identified the common number in both prime factors,
We get,
Therefore, the prime factors common in both \[420\] and \[660\] is,
Given $2 \times 2 \times 3 \times 5 = 60$
Hence the greatest common factor of $420$ and $660$ is $60$.

Note: In the form of multiplying their prime factors, any number can be written. This is referred to as Prime Factorization. In terms of the factor tree, prime variables can also be understood. It also means that the break of a number with all the variables would result in \[0\]remaining. A factor is something that determines the result. To solve this kind of problem, we should understand the basic factors of multiplication and division.