Find the smallest number by which 16384 be divided so that the quotient may be a perfect cube
Answers
Answer:
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
2^{3} \times 2^{3} \times 2^{3} \times 2^{3} \times 2^{2}2
3
×2
3
×2
3
×2
3
×2
2
Now as we can see that there are four 2^{3}2
3
which if multiplied will give a perfect cube but the number multiplied by those four 2^{3}2
3
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 2^32
3
will be perfect cube. Hence if 16384 is divided by 4, then the quotient remaining is \left(2^{3}\right)^{4}=4096.(2
3
)
4
=4096.
Therefore, the smallest number that can be divided to 16384 to give the quotient a perfect cube is 4.
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Step-by-step explanation:
15,384 is the answer for the question