Final answer:
The smallest possible value of k that makes 19845k a perfect cube is 975. This is determined by prime factorizing 19845 and ensuring that all prime factors are present in groups of three to satisfy the conditions of a perfect cube.
Step-by-step explanation:
If the product 19845k is a perfect cube, we must find the smallest positive integer value of k that makes this true. To do that, we first factorize the number 19845. The prime factorization of 19845 is 32 × 5 × 132. A perfect cube is a number that can be written as n3, where n is an integer, meaning all prime factors need to come in triplets for the number to be a perfect cube.
Currently, we have each prime factor of 19845 not in a group of three. To make this number a perfect cube, each prime factor should appear three times. We already have two 3's and two 13's, meaning we need one more 3 and one more 13 to complete their groups of three. We also need two more 5's since there's only one in the factorization. Therefore, the smallest possible value of k which makes 19845k a perfect cube is 3 × 13 × 52 which equals 3 × 13 × 25, or 975.
Therefore, the smallest possible positive integer value of k that makes 19845k a perfect cube is 975.