If (12x)3 = (123)x, then the value of x is

We must solve (12x)₃ = (123)ₓ, noting the meanings of x in each representation.

Key constraints:

• In (12x)₃, x is a digit in base 3, so x must be 0, 1, or 2.

• In (123)ₓ, x is the base and must be greater than the largest digit 3, so x ≥ 4.

Because x would need to be both ≤ 2 and ≥ 4 at the same time, no value can satisfy the constraints. For completeness, we can also solve algebraically:

1. Convert (12x)₃ to decimal: 1×3² + 2×3 + x = 9 + 6 + x = 15 + x.

2. Convert (123)ₓ to decimal: 1×x² + 2×x + 3 = x² + 2x + 3.

3. Set equal: 15 + x = x² + 2x + 3 ⇒ x² + x - 12 = 0 ⇒ (x + 4)(x - 3) = 0.

4. Algebraic roots are x = 3 and x = -4, but neither satisfies the required digit/base constraints.

Conclusion: No valid x exists that meets both roles, so the correct answer is 'None of these'.