Perfect cube adjustment by division: Find the least number by which 175760 should be divided to make it a perfect cube.

Introduction / Context:
To turn a number into a perfect cube, the exponents in its prime factorization must each be multiples of 3. Here, we must divide by the smallest possible factor to achieve that condition.

Given Data / Assumptions:
Number N = 175760.

Concept / Approach:
Factorize N into primes and inspect exponents modulo 3. To get a perfect cube by division, remove (divide out) the minimal prime powers so every exponent becomes a multiple of 3.

Step-by-Step Solution:
Prime factorization: 175760 = 2^4 × 5^1 × 13^3.For a perfect cube, exponents must be 0 mod 3.Adjustments: 2^4 → remove 2^1; 5^1 → remove 5^1; 13^3 already 0 mod 3.Minimum divisor = 2 × 5 = 10.

Verification / Alternative check:
After division: (2^4 × 5 × 13^3) ÷ (2 × 5) = 2^3 × 13^3 = (2 × 13)^3 = 26^3, a perfect cube.

Why Other Options Are Wrong:
5 or 20 or 40 or 100 remove too little or too much, and are not the minimal divisor that achieves a perfect cube.

Common Pitfalls:
Adding factors (multiplication) instead of dividing, or adjusting exponents to multiples of 2 (a square) instead of multiples of 3 (a cube).

Final Answer:
10