Question

If \[N = {12^3} \times {3^4} \times {5^2}\] then find the total number of even factors of N.

Answer 1

Hint: We will first find the total number of factors of the given number with the help of prime factors of the given number. Then in order to find the even factors of N we will subtract the odd factors from the total number of factors. And thus we will get the even number of factors.

Complete step-by-step answer:
Given that
\[N = {12^3} \times {3^4} \times {5^2}\]
Now 3 and 5 are already prime numbers but 12 is not prime. So let’s write it as a product of prime numbers.
\[
  12 = 4 \times 3 = {2^2} \times {3^1} \\
   \Rightarrow N = {({2^2} \times {3^1})^3} \times {3^4} \times {5^2} \\
\]
Yes now all numbers are in prime condition. Now let’s rearrange their powers.
\[
   \Rightarrow N = {({2^2} \times {3^1})^3} \times {3^4} \times {5^2} \\
   \Rightarrow N = {2^{2 \times 3}} \times {3^{1 \times 3}} \times {3^4} \times {5^2}........ \to {\left( {{a^m}} \right)^n} = {a^{m \times n}} \\
   \Rightarrow N = {2^6} \times {3^3} \times {3^4} \times {5^2} \\
\]
We have to add the powers of terms having the same base.
\[
   \Rightarrow N = {2^6} \times {3^{3 + 4}} \times {5^2}......... \to {a^m} \times {a^n} = {a^{m + n}} \\
   \Rightarrow N = {2^6} \times {3^7} \times {5^2} \\
\]
Here we get the final base with their powers. But the question is not solved yet! Now we will find the total number of factors of N.
We have N is in the form \[ \Rightarrow {a^x}{b^y}{c^z}\]
And total numbers of factors are given by \[ \Rightarrow \left( {x + 1} \right)\left( {y + 1} \right)\left( {z + 1} \right)\]
Applying this to the above numbers we get
\[ \Rightarrow \left( {6 + 1} \right)\left( {7 + 1} \right)\left( {2 + 1} \right) = 7 \times 8 \times 3 = 168\]
But we want only the even factors so we have to subtract the odd factors. Now the odd factors are formed by 3 and 5 here. So let’s subtract their factors.
\[ \Rightarrow 168 - 8 \times 3 = 168 - 24 = 144\]
Finally the total number of EVEN factors of N is 144.

Note: In this type of problem of finding the factors of a number, students generally either forget to add 1 to powers of prime numbers during the product or instead of multiplication of powers +1, they add them. But both are wrong. So be careful while solving.