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Write-up


Title
What Values of...?

Problem Statement
For all values of n, where n is a counting number, what values of x make



a whole number?

(Adopted from Mathematics Teacher(1990). March Calendar; Problem 4, p206.)


 

Plans to Solve/Investigate the Problem

Since I am being asked to solve this question for ALL values of n, I know that there need to be many iterations testing numerous possible solutions.  A spreadsheet program is the easiest thing for me to use in solving this problem. 

 

I plan to utilize a formula that references cells containing many values of n (I stopped at n=32 because I am tired and my brain cannot think any higher than that at this time .)

Investigation/Exploration of the Problem

My logical mind (that's an oxymoron) tells me that if I want a whole number result, then my highest possible solution can't be any larger than the smallest result for .  I have found that the smallest result is actually 0 when n = 1, and that gets me nowhere!! 

 

Alas, the next result I get (when I let n = 2) is 30, so this will probably be my upperbound.  Below, you will find one of my iterations.

 
              (n5-n) /x
 
(n5-n) /x
x (n5-n)
0.00000000000000000000000000000 1 30 0
1.00000000000000000000000000000 2 30 30
8.00000000000000000000000000000 3 30 240
34.00000000000000000000000000000 4 30 1020
104.00000000000000000000000000000 5 30 3120
259.00000000000000000000000000000 6 30 7770
560.00000000000000000000000000000 7 30 16800
1092.00000000000000000000000000000 8 30 32760
1968.00000000000000000000000000000 9 30 59040
3333.00000000000000000000000000000 10 30 99990
5368.00000000000000000000000000000 11 30 161040
8294.00000000000000000000000000000 12 30 248820
12376.00000000000000000000000000000 13 30 371280
17927.00000000000000000000000000000 14 30 537810
25312.00000000000000000000000000000 15 30 759360
34952.00000000000000000000000000000 16 30 1048560
47328.00000000000000000000000000000 17 30 1419840
62985.00000000000000000000000000000 18 30 1889550
82536.00000000000000000000000000000 19 30 2476080
106666.00000000000000000000000000000 20 30 3199980
136136.00000000000000000000000000000 21 30 4084080
171787.00000000000000000000000000000 22 30 5153610
214544.00000000000000000000000000000 23 30 6436320
265420.00000000000000000000000000000 24 30 7962600
325520.00000000000000000000000000000 25 30 9765600
396045.00000000000000000000000000000 26 30 11881350
478296.00000000000000000000000000000 27 30 14348880
573678.00000000000000000000000000000 28 30 17210340
683704.00000000000000000000000000000 29 30 20511120
809999.00000000000000000000000000000 30 30 24299970
954304.00000000000000000000000000000 31 30 28629120
1118480.00000000000000000000000000000 32 30 33554400
33333333333300.00000000000000000000000000000 1000 30 1E+15

 

 

Click here to view spreadsheet using Excel. 

 

 

 

Author & Contact

mhouston@rockdale.k12.ga.us