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.
Step-by-step explanation:
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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The largest perfect cube smaller than 16,384 is 15,625. 15,625×1.048576=16,384.
That’s dangerous water, though, as it leads to looking at numbers smaller than 1, allowing your perfect cube quotient to become larger than 16,384, and we head off toward infinity. Negative numbers are something to consider too, since a cube can be negative, unlike a square. So maybe we stick to positive integers for everything.
The perfect cubes smaller than 16,384 are:
15,625 (25³)
13,824 (24³)
12,167 (23³)
10,648 (22³)
9,261 (21³)
8000 (20³)
6,859 (19³)
5,832 (18³)
4,913 (17³)
4,096 (16³) AH-HA!
16,384 ÷ 4,096 = 4, so 16,384 ÷ 4 = 4,096
4,096 is our perfect cube quotient, and the divisor of 16,384 is 4.
Looking at any smaller cubes will lead to larger divisors, so there’s no reason to go on.