SOLUTION: Joe Lucky recently won the New Mexico lottery. The amount of money that he won just happens to be the smallest number of cents (other than 1 cent) that is a perfect square, a perf

Algebra ->  Square-cubic-other-roots -> SOLUTION: Joe Lucky recently won the New Mexico lottery. The amount of money that he won just happens to be the smallest number of cents (other than 1 cent) that is a perfect square, a perf      Log On


   

Question 238440: Joe Lucky recently won the New Mexico lottery. The amount of money that he won just happens to be the smallest number of cents (other than 1 cent) that is a perfect square, a perfect cube, and a perfect fifth power. How much money did he actually win?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The key to this problem is to understand the following property of exponents: %28a%5Ep%29%5Eq+=+a%5E%28%28p%2Aq%29%29:
  • For perfect squares, q = 2: %28a%5Ep%29%5E2+=+a%5E%28%28p%2A2%29%29+=+a%5E%28%282p%29%29. This tells us that any exponent that is a multiple of 2 is a perfect square.
  • For perfect cube, q = 3: %28a%5Ep%29%5E3+=+a%5E%28%28p%2A3%29%29+=+a%5E%28%283p%29%29. This tells us that any exponent that is a multiple of 3 is a perfect cube.
  • For perfect fifth powers, q = 5: %28a%5Ep%29%5E5+=+a%5E%28%28p%2A5%29%29+=+a%5E%28%285p%29%29. This tells us that any exponent that is a multiple of 5 is a perfect fifth power.

So an exponent that is a prefect square, perfect cube and a perfect fifth power, all at the same time, will be a multiple of 2, 3 and 5. The lowest such exponent will be the Lowest Common Multiple (LCM) of these. The LCM of 2, 3 and 5 is 30.

So the smallest number of cents (other than 1) would be 2%5E30+=+1073741824 cents which is $10737418.24.