Final answer:
The smallest positive integer value of k that makes the product 19845k a perfect cube is 975.
Step-by-step explanation:
To find the smallest positive integer value of k that makes the product 19845k a perfect cube, we first need to factorize 19845. After prime factorization, we find that 19845 = 3² × 5 × 13². For a number to be a perfect cube, each prime factor must be raised to a power that is a multiple of 3. In this case, 3² needs another 3, 5 needs two more 5s, and 13² needs another 13. The smallest value of k that accomplishes this is k = 3× 5² × 13 = 3× 25 × 13 = 975. Therefore, the smallest value of k is 975.