Question

By what smallest number should we multiply $8788$ so that the product becomes a perfect cube. Find the cube root of the product.
$A)2,26$
$B)2,6$
$C)22,26$
$D)22,21$

Answer 1

Hint: Since we just need to know such things about the square root numbers and perfect square numbers and perfect cube numbers, A perfect square is the numbers that obtain by multiplying any whole numbers (zero to infinity) twice, or the square of the given numbers yields a whole number like $\sqrt {25} = 5$or $25 = {5^2}$
Similarly, the perfect cube can be expressed in the form of ${3^3} = 27,\sqrt {27} = 3$

Complete step by step answer:
To find the missing number that needs to multiply by the given number to make it as a perfect cube, we use the concept of the prime factorization method.
Prime numbers are the numbers that are divisible by themselves and $1$ only or also known as the numbers whose factors are the given number itself.
But the composite numbers which are divisible by themselves, $1$ and also with some other numbers (at least one number other than $1$ and itself)
Hence the number $8788$ is a composite number and it can be generalized into the form of prime factorization.
Thus we have $8788 = 13 \times 13 \times 13 \times 2 \times 2$ where the number $13$ is repeated three times and hence it is a perfect cube, but the number $2$ repeated two times only and thus $2$ is the smallest number which needs to multiply to get the given number as a perfect cube.
Hence, we have $8788 \times 2 = 13 \times 13 \times 13 \times 2 \times 2 \times 2$
Then we get $17576$ is a perfect cube number.
Now we have to find its cube root, thus we have $\sqrt[3]{{17576}} = 26$ and thus the cube root of the product is $26$
Hence the smallest number is $2$ and cube root is $26$

So, the correct answer is “Option A”.

Note:
We can find the given number is prime or composite by the trial-and-error methods. Divide the number by the prime numbers less than the given number. if the number is divisible exactly by the prime number, it is the composite number, if not then it is the prime number.
The only even prime number is $2$ and all other prime numbers are odd.