Question

Find the smallest number by which 3456 should be divided to make it a perfect cube.

Answer 1

Hint: This type of question is based on the concept of prime factorization. Firstly we will find the prime factor of the number 3456 then we will see that by dividing 3456 by which factor we are getting a number which is a perfect cube.

Complete step by step solution:
It is given in the question that what smallest number when divide the number 3456 yields a perfect cube.
The number given above which has to be divided in order to obtain a perfect cube is as follows:
$3456$
Now, we will use prime factorization method to find the prime factor of the above number:
$\begin{align}
  & 2\left| \!{\underline {\,
  3456 \,}} \right. \\
 & 2\left| \!{\underline {\,
  1728 \,}} \right. \\
 & 2\left| \!{\underline {\,
  864 \,}} \right. \\
 & 2\left| \!{\underline {\,
  432 \,}} \right. \\
 & 2\left| \!{\underline {\,
  216 \,}} \right. \\
 & 2\left| \!{\underline {\,
  108 \,}} \right. \\
 & 2\left| \!{\underline {\,
  54 \,}} \right. \\
 & 3\left| \!{\underline {\,
  27 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
So we got the prime factors as:
$3456=2\times 2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3$
Now as we know we can add the power if the number with same base is multiplied so:
$={{2}^{7}}\times {{3}^{3}}$
As, we can see that power of 3 is divisible by 3 but power of 2 is not divisible by 3 so to make the power of 2 divisible by 3 we will divide the above value by 2 as:
$\begin{align}
  & \Rightarrow \dfrac{{{2}^{7}}\times {{3}^{3}}}{2} \\
 & \Rightarrow {{2}^{6}}\times {{3}^{3}} \\
\end{align}$
Taking cube root we get,
$\begin{align}
  & \Rightarrow \sqrt[3]{{{2}^{6}}\times {{3}^{3}}} \\
 & \Rightarrow {{2}^{2}}\times 3 \\
 & \Rightarrow 12 \\
\end{align}$
So as we can see that by dividing 3456 by 2 we are getting a perfect cube.

Hence, the smallest number which can divide 3456 to make it a perfect cube is 2

Note: Prime Factorization method is used to obtain the prime factors of the number which when multiplied gives the original number. Prime factors are the numbers which are multiplied by 1 and itself so it can’t be factored further and hence is useful. Another way to find the solution is dividing the number by each digit and checking whether a perfect cube is coming or not in this case however the first digit is the answer but in lengthy numbers this method can be difficult and time consuming.