Question

H.C.F of $18$ and $30$ is equal to
A) $6$
B) $5$
C) $4$
D) $3$

Answer 1

Hint: Factors are the set of numbers which are completely divisible by the number. $1$ is the factor of every number. H.C.F stands for Highest Common Factor. H.C.F of the numbers is the multiplication of all the common factors of the numbers.

Complete step-by-step solution:
We have two numbers $18$ and $30$, and H.C.F of the given two numbers is to be determined.
To solve the given question, begin by factoring the two numbers,
First find the factors of $18$. To begin with, start with the smallest prime numbers and divide them by $18$.
Since, $1$ is a factor of every number, so one factor of $18$ is $1$ and the quotient is $18$. So, it can be written as
$18 = 1 \times 18$
Now, check for the smallest prime number $2$. Since $18$ is divisible by $2$ and the quotient comes out to be $9$. So, $2$ is a factor of $18$, and it can be written as
$18 = 1 \times 2 \times 9$
Now, we get the number $9$, again check from the smallest prime $2$, now since $9$ is not divisible by $2$, so we move to the next prime, that is $3$. Since $9$ is divisible by $3$ and the quotient comes out to be $3$, which is prime itself, now our task is complete. So, $3$ is a factor of $9$, and total number of factors of $18$ are given as
$18 = 1 \times 2 \times 3 \times 3$
Now, we will determine the factors of $30$.
Since, $1$ is a factor of every number, so one factor of $30$ is $1$ and the quotient is $30$. So, it can be written as
$30 = 1 \times 30$
Now, check for the smallest prime number $2$. Since $30$ is divisible by $2$ and the quotient comes out to be $15$. So, $2$ is a factor of $30$, and it can be written as
$30 = 1 \times 2 \times 15$
Now, we get the number $15$, again check from the smallest prime $2$, now since $15$ is not divisible by $2$, so we move to the next prime, that is $3$. Since $15$ is divisible by $3$ and the quotient comes out to be $5$, which is prime itself, now our task is complete. So, $3$ and $5$ are factors of $15$, and total number of factors of $30$ are given as
$30 = 1 \times 2 \times 3 \times 5$
So, we have$18 = 1 \times 2 \times 3 \times 3$ and $30 = 1 \times 2 \times 3 \times 5$. From the given factors, it can be seen that the common numbers in the factors are $1 \times 2 \times 3 = 6$. Hence, H.C.F of $30$ and $18$ is $6$.
The correct answer is (A) $6$.

Note: Prime numbers are the numbers that have only two factors, $1$ and the number itself. It can be said that the number is not divisible by any number other than $1$ and the number itself. $1$ is neither a prime number nor composite number. The smallest prime number is $2$. Students have to understand the difference between prime factors and factors to solve such problems.