If (12x)₃ = (123)_x, then the value of x is
(A) 3
(B) 3 or 4
(C) 2
(D) None of these


Answer: (D)

Explanation: Given, (12x)₃ = (123)_x
Since LHS has 3 as the base and RHS has \’x\’ base,

1 * 3*3 + 2 * 3 + x * 1 = 1 * x*x + 2 * x + 3
9 + 6 + x = x2 + 2x + 3
x2 + x - 12 = 0
x2 + 4x - 3x - 12 = 0
x( x + 4 ) - 3(x + 4) = 0
(x + 4)(x - 3) = 0
x = 3, -4 

But, both the values are infeasible.

Alternative explanation –
According to the rules of number systems , the numbers present in a number system should not be greater than the base of the number system.

According to LHS , (12x)₃ tells us that the value of x should be less than 3.
According to RHS , (123)_x tells us that the value of x should be greater than 3 as largest digit in 123 is 3.

Therefore, any combination is not possible.
So, option (D) is correct.

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