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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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Solution solution | Comment 4 of 9 |

2n=a^2
3n=b^3
5n=c^5

So n must be a positive power of 2 multiplied by a positive power of 3 multiplied by a positive power of 5:

n = 2^x * 3^y * 5^z

where

x is odd
x is a multiple of 3
x is a multiple of 5
The smallest such x is 15

y is even
y is 1 less than a multiple of 3
y is a multiple of 5
The smallest such y is 20

z is even
z is a multiple of 3 and therefore of 6
z is 1 less than a multiple of 5
The smallest such z is 24

So n = 2^15 * 3^20 * 5^24

or n = 10^15 * 3^20 * 5^9

The 3^20 * 5^9 factor is 6810125783203125, so n is 6,810,125,783,203,125,000,000,000,000,000

 

 

 


  Posted by Charlie on 2004-11-27 18:51:07
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