How Do Symmetrical Base n and Base m Decimals Relate?

[jmich79]

When fractions F1 and F2 are written as base n decimals, F1=(.7373)base n and F2=(.3737)base n. In base m, F1=(.5252)base m and F2=(.2525) base m. Find (m+n). No trial and error please.

 

[HallsofIvy]

Saying "F1= (.7373) base n" means F1= 7/n+ 3/n²+ 7/n³+ 3/n⁴. Similarly, saying "F1= (.5252) base m" means that F1= 5/m+ 2/m²+ 5/m³+ 2/m⁴. Putting those together you know that 7/n+ 3/n²+ 7/n³+ 3/n⁴= 5/m+ 2/m²+ 5/m³+ 2/m⁴. Likewise, F2= (.3737) base n and F2= (.2525) base m gives 3/n+ 7/n²+ 3/n³+ 7/n⁴= 2/m+ 5/m²+ 2/m³+ 5/m⁴. That gives you two equations to solve for the two unknown numbers m and n.

Edited to replace "y/n³" by "7/n³". My finger slipped!

 

[jmich79]

where are you getting y/n^3 from
I tried doing this method without the y variable and was not able to solve for it. I came up with some crazy nubers. Can you please explain further?

 

[jmich79]

dOES ANYONE KNOW HOW TO SOLVE THIS EQUATION SIMULTANEOUSLY?

 

[Nancarrow]

It seems to me that you're supposed to spot the weird symmetry in the numbers here. '7373', '3737', '5252', '2525'. HallsofIvy outlines a general method of solving this problem when the given numbers have no special pattern, but fk only knows how you'd solve those simultaneous equations!

My thoughts. Try adding F1 and F2 in base m and in base n. Assume, just to make it easier, that n>10 and m>7 (if that's not the case I'm sure it'll turn up in some contradiction or other)


(eta: well, I'm not making much progress... but I still think that it's important to note the symmetry here. It means something, I'm sure of it.)