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Exponential Difficulties 2 (Posted on 2004-11-27) Difficulty: 4 of 5
What's the least positive integer, n, having the following properties:
  • n = (a^2)/2
  • n = (b^3)/3
  • n = (c^5)/5
(where a, b, and c are integers)

See The Solution Submitted by SilverKnight    
Rating: 4.0000 (5 votes)

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Can this be right? | Comment 6 of 9 |

Thoughts…

<o:p> </o:p>

n=(a^2)/2 means 2n = a^2 therefore n must have at least one other factor that is 2.

n=(b^3)/3 means 3n = b^3 therefore n must have at least two other factors that are 3.

n=(c^5)/5 means 5n = c^5 therefore n must have at least four other factors that are 5.

Therefore n will be made up of 2s 3s and 5s.

The number of 2s must be odd (one less than a multiple of 2), and a multiple of 3 and 5.

The number of 3s must be a multiple of 2 and 5, and one less than a multiple of 3.

The number of 5s must be a multiple of 2 and 3, and one less than a multiple of 5.

15 is the lowest number to satisfy the 2s, 20 is the lowest number to satisfy the 3s, and 24 is the lowest number to satisfy the 5s.

Therefore n=(2^15)(3^20)(5^24).

<o:p> </o:p>

That's very big, but I reckon it must be right...


  Posted by Alec on 2004-11-28 11:45:26
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