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 Exponential Difficulties 2 (Posted on 2004-11-27)
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: 2N = a2 3N = b3 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 Solution broll 2016-09-16 00:43:35 There's a better choice FrankM 2008-01-16 00:28:47 answer K Sengupta 2007-08-24 06:03:23 Can this be right? Alec 2004-11-28 11:45:26 Reference Richard 2004-11-28 05:20:54 solution Charlie 2004-11-27 18:51:07 Solution np_rt 2004-11-27 18:43:38 possible solution Bon 2004-11-27 18:29:31 2 out of 3; a start Larry 2004-11-27 16:59:21

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