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 $).