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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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2 out of 3; a start | Comment 1 of 9

Well, n=0 works but it's not positive (unless zero positive is "the least positive")

Otherwise, a must be a power of 2; b a power of 3; c a power of 5

((3*c^5)/5)^(1/3) = b
((2*c^5)/5)^(1/2) = a

Need to find a value for c such that both of the above equations yield integers.

I'm not sure how much this helps, probably not very much, but here are some values where 2 of the 3 original conditions are true (ie 1 of the 2 above equations):

a=12   b=6   n=72
a=96   b=24  n=4608
a=324  b=54   n=52488
a=200  c=10   n=20000
a=6400  c=40  n=20480000
b=1125  c=75  n=474609375
a=48600  c=90  n=1180980000

Edited on November 27, 2004, 5:07 pm

Edited on November 27, 2004, 5:08 pm
  Posted by Larry on 2004-11-27 16:59:21

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