Find the least number by which 175760 must be divided in order to make the quotient a perfect cube.

Introduction / Context:
This arithmetic reasoning question involves prime factorization and properties of perfect cubes. You are given a number, 175760, and asked to find the smallest integer by which it should be divided so that the resulting quotient is a perfect cube. Understanding how exponents of prime factors must behave for a number to be a perfect cube is the key to solving such problems efficiently.

Given Data / Assumptions:

• The original number is 175760.
• We seek the least positive integer k such that 175760 / k is a perfect cube.
• A perfect cube is a number whose prime factor exponents are all multiples of 3.
• We may use prime factorization to understand the structure of 175760.

Concept / Approach:
First, factor 175760 into prime factors and examine the exponents of these primes. For a number to be a cube, each exponent in its prime factorization must be divisible by 3. Since we are dividing by k, we are reducing some of these exponents. Our goal is to remove just enough prime factors so that the remaining exponents become multiples of 3, and the product of the removed factors is as small as possible. We do this by choosing the smallest exponents to subtract that achieve multiples of 3.

Step-by-Step Solution:
Step 1: Prime factorize 175760. It factors as 175760 = 2^4 * 5^1 * 13^3. Step 2: For a perfect cube, all exponents of primes must be multiples of 3. Step 3: Currently, the exponents are: for 2 it is 4, for 5 it is 1, and for 13 it is 3. Step 4: The exponent 3 for 13 is already a multiple of 3, so we do not need to remove any factor of 13. Step 5: For the prime 2, the exponent is 4. To reach the nearest lower multiple of 3, we reduce it to 3 by dividing by 2^1. Step 6: For the prime 5, the exponent is 1. The nearest lower multiple of 3 is 0, so we must remove 5^1 by dividing by 5. Step 7: The smallest number we must divide by is therefore 2^1 * 5^1 = 2 * 5 = 10. Step 8: After division, the quotient is 175760 / 10 = 2^3 * 13^3, which is (2 * 13)^3 = 26^3, a perfect cube.

Verification / Alternative check:
Check that dividing by any smaller factor will not produce a cube. For example, dividing only by 2 gives exponents 3 for 2, 1 for 5, and 3 for 13, so the exponent of 5 is still 1, not a multiple of 3. Dividing only by 5 leaves the exponent of 2 at 4, which is still not a multiple of 3. Thus, you need to remove at least one factor of 2 and one factor of 5, giving the product 10 as the minimal divisor.

Why Other Options Are Wrong:
Option A 6 removes 2 and 3, but 3 is not even a prime factor of 175760, and the exponent of 5 remains unchanged, so the result is not a cube. Option C 9 removes 3^2, again not helpful because 3 is not present in the prime factorization. Option D 30 removes 2, 3, and 5, which is more than necessary and not the least number that works.

Common Pitfalls:
Students sometimes confuse the process for making a number a perfect square with that for a perfect cube and try to pair exponents instead of grouping them in triples. Others attempt to multiply by a number to get a cube instead of dividing, which is a different type of question. Always remember that to make a quotient a perfect cube by division, you must remove primes until all exponents are multiples of 3.

Final Answer:
The least number by which 175760 must be divided to make it a perfect cube is 10.