Question

Let \[N=24500\], then find
(i) The number of ways by which N can be resolved into two factors.
(ii)The number of ways by which 5N can be resolved into two factors.

Answer 1

Hint: First find the number of prime factors of the given number, for example \[N={{x}^{a}}{{y}^{b}}{{z}^{c}}\]. Now, the number of ways in which a given number N can be resolved into two factors will be \[\dfrac{1}{2}(a+1)(b+1)(c+1)\] if N is not a perfect square.

Complete step by step answer:
i) In this part of the question, we have to find the number of ways by which \[N=24500\] can be resolved into two factors. So, here we will find the prime factors of the number as follows:
\[\begin{align}
  & \Rightarrow 24500 \\
 & \Rightarrow 245\times 100 \\
 & \Rightarrow 49\times 5\times 100 \\
 & \Rightarrow {{7}^{2}}\times 5\times {{10}^{2}} \\
 & \Rightarrow {{7}^{2}}\times 5\times {{2}^{2}}\times {{5}^{2}} \\
 & \Rightarrow {{2}^{2}}\times {{5}^{3}}\times {{7}^{2}} \\
\end{align}\]
So the number N is written as \[N={{2}^{2}}\times {{5}^{3}}\times {{7}^{2}}\]. Here the number is resolved in the prime factors form. Next, we know that if the number is of the form \[N={{x}^{a}}{{y}^{b}}{{z}^{c}}\], where x, y and z are the prime factors. Then, the number of ways in which a given number N can be resolved into two factors will be \[\dfrac{1}{2}(a+1)(b+1)(c+1)\], given that N is not a perfect square. Also, here \[N=24500\] is not a perfect square. Now, we have \[a=2,\,\,b=3,c=2\]. So, the number of ways by which \[N=24500\] can be resolved into two factors will be given by:
\[\begin{align}
  & \Rightarrow \dfrac{1}{2}(a+1)(b+1)(c+1) \\
 & \Rightarrow \dfrac{1}{2}(2+1)(3+1)(2+1) \\
 & \Rightarrow 18 \\
\end{align}\]
Finally, we get the required number of ways as 18.

ii) In this part of the question, we have to find the number of ways by which \[5N\] can be resolved into two factors. So, here we will find the prime factors of the number \[5N\], as follows:
\[\begin{align}
  & \Rightarrow 5\times 24500 \\
 & \Rightarrow 5\times 245\times 100 \\
 & \Rightarrow 5\times 49\times 5\times 100 \\
 & \Rightarrow {{7}^{2}}\times {{5}^{2}}\times {{10}^{2}} \\
 & \Rightarrow {{7}^{2}}\times {{5}^{2}}\times {{2}^{2}}\times {{5}^{2}} \\
 & \Rightarrow {{2}^{2}}\times {{5}^{4}}\times {{7}^{2}} \\
\end{align}\]
So the number N is written as \[N={{2}^{2}}\times {{5}^{4}}\times {{7}^{2}}\]. Here, we can see that the number is a perfect square, as we can write the number \[N={{(2\times {{5}^{2}}\times 7)}^{2}}\]. Next, we know that if the number is of the form \[N={{x}^{a}}{{y}^{b}}{{z}^{c}}\], where x, y and z are the prime factors and N is the perfect square. Then, the number of ways in which a given number N can be resolved into two factors will be \[\dfrac{1}{2}\left[ (a+1)(b+1)(c+1)+1 \right]\]. So, now, we have \[a=2,\,\,b=4,c=2\]. So, the number of ways by which \[5N\] can be resolved into two factors will be given by:
\[\begin{align}
  & \Rightarrow \dfrac{1}{2}\left[ (a+1)(b+1)(c+1)+1 \right] \\
 & \Rightarrow \dfrac{1}{2}\left[ (2+1)(4+1)(2+1)+1 \right] \\
 & \Rightarrow 23 \\
\end{align}\]
Finally, we get the required number of ways as 23.

Note: The formula to find the number of ways by which any number can be resolved into two factors should be used carefully because the formula will change if the number is the perfect square.