Calvin visits a Planet called Pandora!

Number Theory Level 2

Calvin went to a newly discovered planet called " Pandora " to research their advancements in the field of Mathematics. He found the following equation scrawled in the dust:

$\large{3x^2 - 25x + 66 = 0 \quad \Longrightarrow \quad x=4 \text{ or } x=9}$

Can you help Calvin in finding the base, which is used in the Number System on Pandora ?

Note : Symbols for digits in the Pandora System and Decimal system have the same value. e.g. 6 in Pandora represents six.

The answer is 17.

2 solutions

Abdeslem Smahi
Aug 11, 2015

Let $a$ be the Number System used in Pandora

Since $x=9$ then $a>9$

so the equation is written in Decimal System as:

$3(x-4)(x-9)=0 \Longrightarrow 3x^2-39x+108=0$

so : $2 \times a+5=39$ and $6 \times a+6=108$

Hence the only solution is $a=17$

Or just 4 9 3=108=6a+6; 6a=102; a=17. No need to double check developing all the equation

damien G - 5 years, 2 months ago

Niranjan Khanderia
Aug 21, 2015

$\text{Since there is 9 in the Pandora, all given integers up to 9 has the same value as}\\ the decimal~ system.\\ So~product~of~roots~=36_{10}=\dfrac{66} 3=22_P.\\ Subtract~2~~34_{10}=20_P=20*P^{1_{10}}.\\ Divide~ both~ by~ 2,~~17_{10}=P^{1_{10}}.\\ \implies~P=17$ $OR$ $Sum~of~roots~=13_{10}=\dfrac{25_P} 3.\\ or~3*13_{10}=25_P.\\ Subtracting~5~from~both,~~39_{10}-5=34_{10}=20_P=2*P^{1_{10}}\\ Divide~ both~ by~ 2,~~17_{10}=P^1_{10}}.\\ \implies~P=17$

Same approach as mine! I used the product of roots ;)

展豪張 - 5 years, 7 months ago

Calvin visits a Planet called Pandora!

The answer is 17.

2 solutions

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