Prime Factorization of 6250
First, find the prime factorization of the number 6250.
$6250 = 625 \times 10$
$6250 = 5^4 \times (2 \times 5)$
$6250 = 2^1 \times 5^5$
Identifying Exponents
The prime factorization is $2^1 \times 5^5$. For a number to be a perfect cube, the exponents of all its prime factors must be multiples of 3.
Calculating the Least Multiplier
Examine the exponents:
- The exponent of 2 is 1. The next multiple of 3 is 3. We need $3 - 1 = 2$ more factors of 2.
- The exponent of 5 is 5. The next multiple of 3 is 6. We need $6 - 5 = 1$ more factor of 5.
Therefore, the least number to multiply by is $2^2 \times 5^1$.
Least multiplier = $4 \times 5 = 20$.
Verification
Multiplying 6250 by 20:
$6250 \times 20 = (2^1 \times 5^5) \times (2^2 \times 5^1)$
$= 2^{(1+2)} \times 5^{(5+1)}$
$= 2^3 \times 5^6$
Since all exponents (3 and 6) are multiples of 3, the result is a perfect cube ($ (2^1 \times 5^2)^3 = 50^3 $).
Similar Questions
- Find the smallest natural number N such that the product $288 \times N$ is a perfect cube.
- The square root of 18769 consists of how many digits?
- The square root of $1\frac{11}{25}$ is:
Evaluate \(21^3 + (-2)^3 + (-19)^3\)
- What is the least positive integer that should be subtracted from 2750, so that the difference is a perfect cube?
- The cube root of 150 will lie between:
- Find the product of square root of 16 and square of 4?
- The value of $\sqrt{142884}$ is:
- The number by which 62557 can be divided to make it a perfect cube is:
- What is the value of $99^3$?
Important Questions from Cube and Cube Root
The cube root of 0.027 is
Two different positions of the same dice are shown, the six face of which are numbered from 1 to 6. Select the number that will be on the face opposite to the face having the number '3'.
What is the cube root of 1728?
If 5 \(\sqrt{3}\) + \(\sqrt{75}\) = 17.32, then the value of 14 \(\sqrt{3}\) + \(\sqrt{108}\) is:
Two orientations of a dice are shown. This dice can be obtained by folding which of the option figures along the lines?
