Question

, The equation 166⇥56 = 8590 is valid in some base b " 10 (that is, 1, 6, 5, 8, 9, 0 are digits in base

b in the above equation). Find the sum of all possible values of b " 10 satisfying the equation.​

Answer 1

Answer:

Possible value of b = 12

Step-by-step explanation:

(166)b  * (56)b   = (8590)b

(b²*1 + b*6 + 6) * (b*5 + 6) = (b³*8 + b²*5 + b*9 + 0)

=> (b² + 6b + 6) (5b + 6) = 8b³ + 5b² + 9b

=> 5b³ + 36b² + 66b + 36 = 8b³ + 5b² + 9b

=> 3b³ - 31b² -57b -36 = 0

=> 3b³ - 36b² + 5b² - 60b + 3b - 36 =0

=> 3b²(b - 12) + 5b(b - 12) + 3(b-12) =0

=>(3b² + 5b + 3)(b-12) = 0

3b² + 5b + 3 will give imaginary roots

=> b - 12 = 0

=> b = 12

Possible value of b = 12

Even if we consider all values including imaginary

Sum of all roots  = -(-31)/3 = 31/3