Understand the Goal

The task is to determine the specific number that divides $62557$ to yield a perfect cube.

Prime Factorization of 62557

To find the required divisor, we first find the prime factorization of $62557$.

• Divide $62557$ by potential prime factors. Testing known factors reveals:
• $62557 = 47 \times 1331$
• We know that $1331$ is the cube of $11$, i.e., $1331 = 11^3$.
• Thus, the prime factorization of $62557$ is $47^1 \times 11^3$.

Identify Factors for Perfect Cube Requirement

A number is a perfect cube if all exponents in its prime factorization are multiples of $3$.

• In $47^1 \times 11^3$:
• The exponent for $11$ is $3$, which is a multiple of $3$.
• The exponent for $47$ is $1$, which is not a multiple of $3$.

Determine the Divisor

To transform $62557$ into a perfect cube through division, we must divide by the prime factors whose exponents are not multiples of $3$.

We need to remove the factor $47^1$. This is achieved by dividing by $47$.

Performing the division:

$ \frac{62557}{47} = \frac{47^1 \times 11^3}{47^1} = 11^3 $

The result, $11^3$, is a perfect cube ($1331$).

Conclusion

The number required to divide $62557$ to make it a perfect cube is 47.