Find the smallest positive integer by which 16200 must be multiplied so that the resulting product becomes a perfect cube.

Introduction / Context:
This question checks understanding of prime factorization and the conditions required for a number to be a perfect cube. In aptitude and number theory problems, converting numbers into their prime factors is a powerful technique, especially when determining whether they are perfect squares, cubes, or higher powers. Here, we are given 16200 and asked for the least integer that, when multiplied with it, turns the product into a perfect cube. That means we want all prime exponents in the factorization to be multiples of 3.

Given Data / Assumptions:

• Given number: 16200.
• We need to multiply 16200 by a positive integer k.
• The product 16200 * k must be a perfect cube.
• A perfect cube has all prime factor exponents as multiples of 3.

Concept / Approach:
We first factorize 16200 into its prime factors. After obtaining the exponents of each prime, we check which exponents are not multiples of 3. For each such prime, we find the smallest additional power needed to make the exponent a multiple of 3. The required multiplier k is then formed by multiplying these extra prime factors together. This ensures that the combined exponent for each prime in 16200 * k is divisible by 3, making the product a perfect cube.

Step-by-Step Solution:
Step 1: Factorize 16200. Step 2: Note that 16200 = 162 * 100. Step 3: Factorize 162 as 2 * 81 = 2 * 3^4. Step 4: Factorize 100 as 2^2 * 5^2. Step 5: Combine the factors: 16200 = 2^3 * 3^4 * 5^2. Step 6: For a perfect cube, the exponents of all primes must be multiples of 3. Step 7: The exponent of 2 is 3, already a multiple of 3. Step 8: The exponent of 3 is 4. To reach the next multiple of 3 (which is 6), we need 2 more factors of 3, that is 3^2. Step 9: The exponent of 5 is 2. To reach the next multiple of 3 (which is 3), we need one more factor of 5, that is 5^1. Step 10: Therefore, the required multiplier k is 3^2 * 5 = 9 * 5 = 45.

Verification / Alternative check:
Multiply 16200 by 45: the combined factorization is 2^3 * 3^4 * 5^2 * 3^2 * 5^1 = 2^3 * 3^6 * 5^3. Now each exponent is a multiple of 3: 3 for base 2, 6 for base 3, and 3 for base 5. Therefore, the product is indeed a perfect cube of the form (2 * 3^2 * 5)^3. This confirms that 45 is the correct smallest multiplier.

Why Other Options Are Wrong:
Option (b) 48: Its factors do not fix all exponents to multiples of 3 with the smallest possible product, and it is larger than necessary.
Option (c) 360: This is much larger and not the least multiplier, even though it may also yield a perfect cube.
Option (d) 36: This changes the exponents but does not make all of them multiples of 3, so the product is not a perfect cube.

Common Pitfalls:
Common errors include incorrect prime factorization or forgetting that exponents must be multiples of 3 for a cube. Some learners also choose a multiplier that works but is not the smallest possible one. Focusing on each prime factor exponent and adjusting it to the nearest higher multiple of 3 helps avoid mistakes and ensures minimality.

Final Answer:
The least integer that must be multiplied with 16200 is 45.