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A sixth power in many ways (Posted on 2018-02-22) Difficulty: 3 of 5
How many different possible expressions are there for m^n if:

(i) both m and n are positive integers,not necessary distinct,
(ii) both numbers are below 10000 and
(iii) m^n can be written as k^6, k an integer?

Explain your result.

No Solution Yet Submitted by Ady TZIDON    
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Solution solution Comment 1 of 1
m^n is a perfect sixth power if:

n is a multiple of 6
m is a perfect square and n is a multiple of 3
m is a perfect cube and n is even
m is itself a perfect sixth power

As these conditions are not mutually exclusive, the hard part is subtracting out duplicates.

For m:

There are 99 positive (non-zero) perfect squares below 10000.
There are 21 perfect cubes below 10000.
There are 4 that are both perfect squares and perfect cubes, as they are perfect 6th powers.

For n:

There are 3333 that are a multiple of 3.
There are 4999 that are even. (remember 10000 is not included)
There are 1666 that are both (i.e., a multiple of 6).

We'll separate out the mutually exclusive cases of m:

Cases:

m a perfect square but not a perfect cube: 99 - 4 = 95

95 * 3333 = 316635 as any multiple of 3 for n will work

m a perfect cube but not a perfect square: 21 - 4 = 17

17 * 4999 = 84983 as any multiple of 2 for n will work

m a perfect 6th power: 4

4 * 9999 = 39996 as any value of n will work

m neither a perfect square nor a perfect cube: 9999 - 99 - 21 + 4 = 9883

9883 * 1666 = 16465078 as n needs to be a multiple of 6

Adding them all up:
                                                                           
    316635                                                                        
     84983                                                                         
     39996                                                                         
  16465078 
  --------
  16906692                                                                      


  Posted by Charlie on 2018-02-22 11:12:10
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