Question

Given (135)_{base x} + (144)_basex = (323)_basex

What is the value of base x?

1. 5
2. 3
3. 12
4. 6

Answer 1

Correct Answer - Option 4 : 6

Concept:

To convert any number which is in a different base to a decimal number system, we use binary-weighted representation.

Example: Let the number be ( abc⋯ ⋯ yz)r

Now to convert the above number into the decimal number system, we write:

a × rn-1 + b × rn-2 + ⋯ ⋯ + y × r1 + z × r0

Application:

(135)x + (144)x = (323)x

If the two numbers are equivalent, they will remain equal in other number systems as well.

Converting both the sides to decimal and equating, we get:

(1 × x2 + 3 × x1 + 5 × x0) + (1 × x2 + 4 × x1 + 4 × x0) = 3 × x2 + 2 × x1 + 3 × x0

x² + 3x + 5 + x² + 4x + 4 = 3x² + 2x + 3

x² - 5x - 6 = 0

x² + x - 6x - 6 = 0

x (x + 1) - 6 (x + 1) = 0

(x - 6) (x + 1) = 0

x = 6 (Valid)