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 $).
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Important Questions from Cube and Cube Root
Find the value of (25)3 + (-29)3 + (4)3
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