Question

Is 900 a perfect cube?
(A) No
(B) Yes
(C) None of these
(D) Can’t say

Answer 1

Hint: We are given a number and we are asked to check whether the given number is a perfect cube or not. We will begin by computing or finding the prime factorization of the given number. So, we get the prime factorization as follows, \[900={{2}^{2}}\times {{3}^{2}}\times {{5}^{2}}\]. We know a perfect cube number has the number 3 or its multiples as the power of the prime factors. Hence, we will have the answer whether the given number is a perfect cube or not.

Complete step-by-step solution:
According to the given question, we are given a number and we have to check if the given number is a perfect cube or not and select the most appropriate answer from the given option.
 The number given to us is,
\[900\]
Firstly, we will have to find the factors of the given number and for that we can use the prime factorization method. Prime factorization involves writing the given number as a product of prime numbers.
We will divide the number by the least prime number and continue dividing the number until all the factors are prime numbers only, so we have,
\[\begin{align}
  & 2\left| \!{\underline {\,
  900 \,}} \right. \\
 & 2\left| \!{\underline {\,
  450 \,}} \right. \\
 & 3\left| \!{\underline {\,
  225 \,}} \right. \\
 & 3\left| \!{\underline {\,
  75 \,}} \right. \\
 & 5\left| \!{\underline {\,
  25 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
Now, we will write the factors of 900 as the product of its prime factors, we get,
\[900={{2}^{2}}\times {{3}^{2}}\times {{5}^{2}}\]
But, we know that a perfect cube has its prime factors with the power as 3 or its multiples.
Hence, 900 is not a perfect cube.
Therefore, the correct option is (A) No.

Note: The perfect cube will always have the power of its factors as 3 or its multiples. For example – 8 is a perfect cube as it has the prime factors as, \[8=2\times 2\times 2={{2}^{3}}\]. So, here we can see that the power of the prime factor is 3 and so 8 is a perfect cube. Another example is 64, its prime factorization is, \[64=2\times 2\times 2\times 2\times 2\times 2={{2}^{6}}\] and here too we can see that the power of the prime factor is a multiple of 3, hence, 64 is a perfect cube.