Question

Find the base r if (159)10 =(186) r​

Answer 1

Answer:

In general, since we are comfortable dealing with Decimal system, the best way to go about under the circumstances is to convert one Number into decimal Number and then with the base x, convert another Number into Decimal Number and equate them to find the value of x.

So, (152)8 is 64*1+8*5+2, which is 106 in Decimal System.

(211)x is 2x^2+ 1*x+1 in Decimal System.

Now, we solve 2x^2+x+1 = 106

i. e. 2x^2+x-105=0

Therefore 2x^2–14x+15x-105=0

Therefore 2x(x-7)+15(x-7)=0

Therefore (x-7)(2x+15)=0

Therefore either x = 7 or x = -15/2.

Obviously x = -15/2 isnot the answer. So, x = 7 is the answer, which can be very easily verified. 2*7^2+1*7+1 = 106 which issame as (152)8.

Thus the base for 211 is 7.