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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Binary Numbers

Olympiad workbook · IOQM · Binary Numbers

The equation 166 × 56=8590 is valid in some base b ≥ 10 (that is...

The equation 166×56=8590166 \times 56=8590 is valid in some base b≥10b \geq 10 (that is 1,6,5,8,9,01,6,5,8,9,0 are digits in base bb in the above equation). Find the sum of all possible values of b≥10b \geq 10 satisfying the equation.

Correct answer

12

Step-by-step solution

166=b2+6b+656=5b+68590=8b3+5b2+9b Now, (b2+6b+6)(5b+6)=8b3+5b2+9b⇒5b3+36b2+66b+36=8b3+5b2+9b⇒3b3−31b2−57b−36=0⇒(b−12)(3b2+5b+3)=0⇒b=12\begin{aligned} & 166=b^2+6 b+6 \\ & 56=5 b+6 \\ & 8590=8 b^3+5 b^2+9 b \\ & \text { Now, } \left(b^2+6 b+6\right)(5 b+6)=8 b^3+5 b^2+9 b \\ & \Rightarrow 5 b^3+36 b^2+66 b+36=8 b^3+5 b^2+9 b \\ & \Rightarrow 3 b^3-31 b^2-57 b-36=0 \\ & \Rightarrow (b-12)\left(3 b^2+5 b+3\right)=0 \\ & \Rightarrow b=12 \end{aligned} We have only one bb which is 12 So, sum =12=12
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