What is the least number that 500 must be multiplied by to make it a perfect cube?

Answer

Hint: First we have to define what the terms we need to solve the problem are.
First of all, we just need to know such things about the perfect numbers, which 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 $\sqrt 4 = 2$ or $4 = {2^2}$ is the perfect square; similarly for the perfect cube we can define, ${3^3} = 27$ or $\sqrt[3]{{27}} = 3$ which is the perfect cubes example;

Complete step-by-step solution:
Since the given question is to find the smallest number that will surely need to multiplied to get the perfect cube as defined above is $2916$ so this number is not the perfect cube since it is not well completed and hence, we need another number to be multiplied and get the resultant;
Let $2916$ can be written as in the form of multiplication of the twos and threes like
$3 \times 3 \times 3 = 27$ Again $3 \times 3 \times 3 = 27$ and also $2 \times 2 = 4$ so if we multiplied the all terms, we get
$27 \times 27 \times 4 = 2916$ That means we can able to reform the number $2916$ as in the form of
$3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 2 \times 2 = 2916$ as we can see there are a total six three and two twos only (perfect square). Now six three can be assumed to be the perfect cube (since first three \[3\] is perfect and last three \[3\] are perfect cube) and then there are only two \[2s\] so we need another $2$ to get that number as a perfect cube; because after multiplied by two only we get three twos and then only it will be perfect cube;
Hence $2$ is the smallest number that will need to multiply by $2916$ to get that perfect square.

Note: Since if two is multiplied we get the number as $2916 \times 2 = 5832$ and then we will separate like above we did we get; $3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 2 \times 2 \times 2 = 5832$ hence all the twos are perfect square by three terms and all threes are perfect square by six terms thus the resultant.

Question:

Cheryl E.

Algebra

9 months, 2 weeks ago

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what is the smallest number by which 13500 should be multiplied so that the product is a perfect cube

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Video Transcript

What is the smallest number by which 13500 should be added to make a perfect cube? As the given number is 13500 point, we have to find the smallest number we can to make a perfect cube. Factoring this is the first thing we will do. It is understood that to be a perfect cube, there will be a pair of factors by 3, so factor 13500 will be 2 raised to the power to multiply with 5 raised to the power 3 and 3 raised to the power 3. We have to add 2 to make them a perfect cube because there is not a perfect pair of 3 or 2. It will be 2 raised to the power 3, 5 raised to the power 3, and 3 raised to the power 3. It will be a perfect cube, so we can say that the smallest number, by which we have to add up, is 2.

Solution:

(i) 243

Prime factors of 243 =

Here 3 do not appear in 3’s group.

Therefore, 243 must be multiplied by 3 to make it a perfect cube.

(ii) 256

Prime factors of 256 = 2\times2\times2\times2\times2\times2\times2\times2

Here one factor 2 is required to make a 3’s group.

Therefore, 256 must be multiplied by 2 to make it a perfect cube.

(iii) 72

Prime factors of 72 = 2\times2\times2\times3\times3

Here 3 does not appear in 3’s group.

Therefore, 72 must be multiplied by 3 to make it a perfect cube.

(iv) 675

Prime factors of 675 = 3\times3\times3\times5\times5

Here factor 5 does not appear in 3’s group.

Therefore 675 must be multiplied by 3 to make it a perfect cube.

(v) 100

Prime factors of 100 = 2\times2\times5\times5

Here factor 2 and 5 both do not appear in 3’s group.

Therefore 100 must be multiplied by 2\times5= 10 to make it a perfect cube.

What is the perfect cube of 500?

What is the smallest number by which 500 may be multiplied to get a perfect square?

What is the smallest number that can be multiplied by 500?

IS 500 a perfect cube state with reason?