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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)

  Submitted by SilverKnight    
Rating: 4.0000 (5 votes)
Solution: (Hide)
Let N = (2^i)(3^j)(5^k) be such a number. Then there must exist numbers a, b, c so that:
  1. 2N = a2
  2. 3N = b3
  3. 5N = c5
The exponent i must be odd from (1), a multiple of 3 from (2) and a multiple of 5 from (3). The smallest exponent which meets these requirements is 15.

The exponent j must be even from (1), one less than a multiple of 3 from (2), and a multiple of 5 from (3). The smallest exponent which meets these requirements is 20.

The exponent k must be even from (1), a multiple of 3 from (2), and one less than a multiple of 5 from (3). The smallest exponent which meets these requirements is 24.

Therefore, the smallest such integer is N = (2^15)(3^20)(5^24) = 6,810,125,783,203,125,000,000,000,000,000.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSolutionbroll2016-09-16 00:43:35
SolutionThere's a better choiceFrankM2008-01-16 00:28:47
answerK Sengupta2007-08-24 06:03:23
Can this be right?Alec2004-11-28 11:45:26
ReferenceRichard2004-11-28 05:20:54
SolutionsolutionCharlie2004-11-27 18:51:07
Solutionnp_rt2004-11-27 18:43:38
possible solutionBon2004-11-27 18:29:31
2 out of 3; a startLarry2004-11-27 16:59:21
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