Question

How do you find the GCF of $3$ and $9$ ? \[\]

Answer 1

Hint: We recall dividend, divisor, quotient and factor from arithmetic division. We recall that factor is a divisor that exactly divides a dividend and the greatest among the factors of two numbers is called greatest common factor (GCF). We find the common factors of 3 and see which is the greatest of them. \[\]

Complete step by step answer:
We know that in arithmetic operation of division the number we are going to divide is called dividend, the number by which divides the dividend is called divisor. We get a quotient which is the number of times the divisor is of dividend and also remainder obtained at the end of division. If the number is $n$, the divisor is $d$, the quotient is $q$ and the remainder is $r$, they are related by the following equation,
\[n=dq+r\]
Here the divisor can never be zero. If the remainder $r=0$ is zero we have
\[n=dq\]
 Here we say $d$ exactly divides $n$ and $d,q$ are factors of $n$. If $d$ exactly divides two numbers say $m,n$ then we say $d$ is a common factor of $m,n$. The greatest of such factors like $d$ is called greatest common factor (GCF). \[\]
We are asked to find the GCF of 3 and 9. The factors of 3 are 1 and 3. The factors of 9 are $1,3,9$. So $3$ is the only factor that is common to both 3 and 9. Since there are no other factors it is also the greatest common factor(GCF). So we have;

\[\text{GCF}\left( 3,9 \right)=3\]

Note: We note that If $d$ exactly divides $n$ then we say $n$ is a multiple of $d$. If ${{d}_{1}},{{d}_{2}}$ both exactly divide $n$ then $n$ is a common multiple of ${{d}_{1}},{{d}_{1}}$.If $n$ is the smallest of common multiples of ${{d}_{1}},{{d}_{1}}$ then we say $n$ is the least common multiple(LCM) of ${{d}_{1}},{{d}_{2}}$. Here $\operatorname{LCM}\left( 3,9 \right)=3$. If we are given two numbers where one is a factor of another then a smaller number is GCF and a greater number is LCM like in this problem.