Question

The number of different factors of 3003 is
(a) 2
(b) 15
(c) 7
(d) 16

Answer 1

Hint: To find the number of different factors of 3003, we have to find its prime factors using the ladder method or factor tree. Then, we have to write the prime factorization in the exponential form. We will use the equation for determining the number of factors which is given by $d\left( n \right)=\left( a+1 \right)\left( b+1 \right)\left( c+1 \right)+...$ , where $d\left( n \right)$ is the number of divisors of the number, a, b, c .. are the exponents of the prime factors of the number. we have to substitute the values in this equation and simplify.

Complete step by step answer:
We have to find the number of different factors of 3003. Let us find the prime factors of 3003. We will use the ladder method to do this.
$\begin{align}
  & \text{ }3\left| \!{\underline {\,
  3003 \,}} \right. \\
 & \text{ }7\left| \!{\underline {\,
  1001 \,}} \right. \\
 & 11\left| \!{\underline {\,
  143 \,}} \right. \\
 & 13\left| \!{\underline {\,
  13 \,}} \right. \\
 & \begin{matrix}
   {} & \text{ }\text{ }1 \\
\end{matrix} \\
\end{align}$
Therefore, the prime factorization of $3003=3\times 7\times 11\times 13$ .
Let us write the prime factorization in the exponential form.
$3003={{3}^{1}}\times {{7}^{1}}\times {{11}^{1}}\times {{13}^{1}}$
We know that the equation for determining the number of factors (or divisors) is given by
$d\left( n \right)=\left( a+1 \right)\left( b+1 \right)\left( c+1 \right)...$
where $d\left( n \right)$ is the number of divisors of the number, a, b,c .. are the exponents of the prime factors of the number. We have to find $d\left( 3003 \right)$ .
$\begin{align}
  & \Rightarrow d\left( 3003 \right)=\left( 1+1 \right)\times \left( 1+1 \right)\times \left( 1+1 \right)\times \left( 1+1 \right) \\
 & \Rightarrow d\left( 3003 \right)=2\times 2\times 2\times 2 \\
 & \Rightarrow d\left( 3003 \right)=16 \\
\end{align}$
Therefore, 3003 has 16 factors.

So, the correct answer is “Option d”.

Note: Students can also use the factor tree method to find the prime factors. They must understand the parameters of the equation for d(n) as there is a high chance of making a mistake by substituting the prime factors as a, b, c,… instead of their exponents.