SOLUTION: What is the smallest positive integer n for which 45n is a perfect cube of an integer?

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Question 319260: What is the smallest positive integer n for which 45n is a perfect cube of an integer?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First, factor 45n into 3*3*5*n. In order for 3*3*5*n to be a perfect cube, each prime factor must come in sets of triples. So we're missing one 3 and two 5 factors which means that n=3*5*5=75


So the answer is n=75 making the final number to be 45*75=3375


Using a calculator, we find that root%283%2C3375%29=15. We could also notice that since 3*3*3*5*5*5=3375, we can just rearrange the terms to get 3%5E3%2A5%5E3=%283%2A5%29%5E3=15%5E3=3375 which would mean that 15%5E3=3375 (ie showing that 3375 is a perfect cube)