If 23x + 101x = 130x, find the value of x

Interpreting the question: the numbers are written in base x, i.e. 23ₓ + 101ₓ = 130ₓ.

Step 1: Convert each number to base 10

• 23ₓ = 2x + 3
• 101ₓ = x² + 0·x + 1 = x² + 1
• 130ₓ = x² + 3x + 0 = x² + 3x

Step 2: Form the equation

(2x + 3) + (x² + 1) = x² + 3x

x² + 2x + 4 = x² + 3x

Step 3: Solve

Subtract x² from both sides:

2x + 4 = 3x

x = 4

Step 4: Check

Base 4: the digits used (0, 1, 2, 3) are all valid, since each digit must be less than the base.

• 23₄ = 2(4) + 3 = 11
• 101₄ = 16 + 1 = 17
• 130₄ = 16 + 12 = 28

11 + 17 = 28 ✓

Answer: D (x = 4)

Tip: In base questions, always convert each number to base 10 using place values (the rightmost digit is x⁰, then x¹, x², and so on). The x² terms often cancel, leaving a simple linear equation. Also remember the base must be bigger than the largest digit in the numbers. Here the digit 3 means x ≥ 4, which fits perfectly.