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SIX=380, SEVEN=36467, EIGHT=68921
First, Make a list of possible numbers for EIGHT and for SIX:
EIGHT = 10648, 13824, 19683, 24389, 32768, 42875, 54872, 68921
SIX = 132, 156, 182, 210, 240, 306, 342, 380, 420, 462, 506, 650, 702, 756, 812, 870, 930
Now find pairs of {SIX,EIGHT} which have only the second digit(I) the same.
{SIX,EIGHT} = {702,10648}, {930,13824}, {420,32768}, {380,68921}
If {SIX,EIGHT} = {702,10648} then SEVEN = 71315, 71319, 71513, 71519, 71913, or 71915.
71315 = 5*14263
71319 = 3*23773
71513 = 13*5501
71519 = 7*10217
71913 = 3*23971
71915 = 5*14383
{SIX,EIGHT} = {702,10648} is not a solution
If {SIX,EIGHT} = {930,13824} then SEVEN = 91516, 91517, 91615, 91617, 91715, or 91716.
91516 = 2*45758
91517 = 23*3979
91615 = 5*18323
91617 = 3*30539
91715 = 5*18343
91716 = 2*45858
{SIX,EIGHT} = {930,13824} is not a solution
If {SIX,EIGHT} = {420,32768} then SEVEN = 43135, 43139, 43531, 43539, 43931, or 43935.
43135 = 5*8627
43139 = 179*241
43531 = 101*431
43539 = 3*14513
43931 = 197*223
43935 = 5*8787
{SIX,EIGHT} = {420,32768} is not a solution
If {SIX,EIGHT} = {380,68921} then SEVEN = 36465, 36467, 36564, 36567, 36764, or 36765.
36465 = 5*7293
36467 is prime
36564 = 2*18282
36567 = 3*12189
36764 = 2*18382
36765 = 5*7353
Since 36467 is the only prime, the only solution is SIX=380, SEVEN=36467, and EIGHT=68921. |
