4 is the smallest number in which 16384 can be divided so that the quotient may be a perfect cube.
Given:
16384
To find:
Find the smallest number by which 16384 can be divided so that the quotient may be perfect cube.
Solution:
The dividend of the question is 16384
The divisor of the question is X
The property of the quotient is that it is a perfect square.
Thereby, let start by taking out the prime factorization of 16384 which is
Now as we can see that there are four which if multiplied will give a perfect cube but the number multiplied by those four is 4. 4 is the only number which is not a cube there by if we take out 4 from the factorization then the product of will be perfect cube. Hence if 16384 is divided by 4, then the quotient remaining is
Therefore, the smallest number that can be divided to 16384 to give the quotient a perfect cube is 4.
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Answer:
The required number is 2.
Step-by-step explanation:
The given number is 16384.
The prime factorization of this number is:
[tex]16384 = 2\times2 \times2\times2\times2\times2\times2\times 2\times2\times 2\times2\times2\times 2\times2[/tex]
[tex]16384 = 2^{14}[/tex]
Making pair of 3 2's, we get such 4 pairs. We are left with 2 × 2
Thus, we need to multiply the given number by 2 to make it a perfect cube.
[tex]16384 \times 2 = 32768 = (32)^{3}[/tex]
Cube root of the new obtained number is 32.
