There's a similar question on here already but I don't think the answers are applicable in this case.
Question: Find the smallest three consecutive integers for which the first integer is divisible by the square of a prime; the second integer by the cube of a prime; and the third integer by the fourth power of a prime.
My attempt: Let the three consecutive integers be a, b and c. Then
where $d,e,f$ are prime. This is equivalent to
Then after applying the Chinese Remainder Theorem,
I don't think this is getting me anywhere. Is this sort of general approach going to work or am I wasting my time?