Question

By which smallest number 8788 must be divided so that quotient is a perfect cube?

Answer 1

Hint: A perfect cube is a number that can written as power of 3 of any number, for example let us assume a number 8 then it can be written as \[{2^3}\] so the 2 is the perfect cube of the number 8. In the given question as a first step, find the prime factorization of the number and observe the output to decide the divisor.

Complete step-by-step answer:
Given:
The dividend is $d = 8788$.
The prime factorization of the 8788 is as follows.
Firstly taking 2 as factor,
 \[\Rightarrow 8788 = 2 \times 4394\]
Again taking 2 as factor,
\[\Rightarrow 8788 = 2 \times 2 \times 2187\]
Now, taking 13 as the factor, then
\[\Rightarrow 8788 = 2 \times 2 \times 13 \times 169\]
Again taking 13 as the factor, then
 \[\Rightarrow 8788 = 2 \times 2 \times 13 \times 13 \times 13\]
Therefore, the prime factorization will be written as
\[\Rightarrow {2^2} \times {13^3}\],
Here 13 is repeated 3 times and 2 is just two times, so we can make a perfect cube by 13, so to make that we have to eliminate the numbers other than 13 in the prime factorization. So the smallest number that can be eliminated will be
\[\Rightarrow 2 \times 2 = 4\].
Dividing 8788 by 4 to find the quotient that can be a perfect cube.
\[\Rightarrow \dfrac{{8788}}{4} = 2197\], so 4 will be the smallest number and quotient that that can be a perfect cube is 2197

Note: Here be sure about the factorization, our aim is to bring a perfect cube for the given number, so the factorization should be such a way that we can easily decide the number to divide with 8788. Check the quotient again after division, whether it is satisfying the requirement of a perfect cube.