An Out of this World Problem!

Number Theory Level 2

The first expedition to a distant planet found only the ruins of a civilization. The explorers were able to translate the "extraterrestrial" equation they found there as follows:

$5x^2 -50x +125 = 0. \text{ Therefore, } x \text{ is equal to } 5 \text{ or } 8.$

This was an extremely strange result. The value $x=5$ seemed satisfactory, but $x=8$ required some explanation. If their number system were similar to ours, how many fingers would you say the inhabitants in that planet had?

Make the assumption that if the inhabitants had $n$ fingers, then they work in base $n$ .

The answer is 13.

6 solutions

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Bhaskar Sukulbrahman
Apr 17, 2014

Sum of roots = ( $5\times b$ )/5 = b = 8+5=13

where ( $5\times b$ ) is 50 in base 10.

Wow, that's very nice. I just bashed it out like most other people.

Calvin Lin Staff - 7 years, 1 month ago

Thank you. But actually I did it in another way, then I found this easier solution so I think I should share it. :-)

Bhaskar Sukulbrahman - 7 years, 1 month ago

That's exactly how I solved it, like Bhaskar. Trickzzzzz... :D

Finn Hulse - 7 years, 1 month ago

Since 5 and 8 are solutions 5(x-5)(x-8)=5x^2-5.(5+8)+5.5.8 which gives 8+5=10 which gives base 13 and 125 to the base 13 is 200 (169+26+5) which confirms the answer. I took quite some time but this is my first. I dont quite follow Bhaskar's brief solution perhaps he went to a different school anyway congrats.

Ramesh V - 7 years, 1 month ago

My solution is based on this property of quadratic equations:

For a quadratic equation A $x^{2}$ +B $x$ +C=0

Sum of roots, $x1$ + $x2$ = - $\frac{B}{A}$ .

Product of roots, $x1 \times x2$ = $\frac{C}{A}$ .

I assumed that everyone must know it, so I didn't explain this part.

Bhaskar Sukulbrahman - 7 years, 1 month ago

Thanks

Ramesh V - 7 years, 1 month ago

Yeah.

Finn Hulse - 7 years, 1 month ago

I did the trial and error way... I solved it fro base-9 'till 13! :)

lynn saibor - 7 years, 1 month ago

Good thinking - I always forget about the -b/a, c/a.. etc. especially when solving simple stuff like this!

Rohit Gupta - 7 years, 1 month ago

Mayank Chaurasia
Apr 20, 2014

5(x-5)(x-8)=0 5x^2 - 65x + 200 = 0 So 65 in the decimal system is 50 in an n'th system. 65/5=13, so n=13

Adrian Neacșu
Apr 17, 2014

Write the numbers in base n and use the fact that 8 is a solution.

5(x-5)(x-8)=0 5x^2 - 65x + 200 = 0 So 65 in the decimal system is 50 in an n'th system. 65/5=13, so n=13

Wouter Fokkema - 7 years, 1 month ago

Daryl Yara
Apr 24, 2014

Ok so my solution is a really simple logic.... If they have the same number system as us but with 5 or 8 works the same way as if 8+8=10 as 5+5=10, then maybe 5+8=10, too so 8 is treated as if it's 5. Also, I sort of guessed this answer since I just pictured the idea of an alien having 8 fingers in its left hand and 5 on the other, so they're pretty messed up on their number systems. Isn't that cute? :D xD

Daniel Magen
Apr 18, 2014

if 5 and 8 are the solutions then the formula must be in the form of (in base 10) (x-5)(x-8) = x^2 - 13x + 40. multiply it by 5 and you reach the formula 5x^2 -65+200=0. you can compare 65 in base 10 to 50 in base n => 65 = 5*n => n=13

nice

Amit Chopra - 7 years, 1 month ago

Cesar Conterno
Apr 26, 2014

transforming the equation in decimal numerical basis, we have: number of fingers: y ; 5x^2 - 5yx + y^2 + 2y + 5 = 0; x=5 and x=8 are solution, so y=13.

just do not know if it's in both hands or in each hand

cesar conterno - 7 years, 1 month ago

An Out of this World Problem!

The answer is 13.

6 solutions

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