If 23$_y$ = 1111$_{\text{two}}$, find the value of y

Analyze the question: We have a number 23 in an unknown base y, which equals 1111 in base two. We need to find y.

Concept: To compare numbers in different bases, convert both to base 10.

• A number 23 in base y = 2y + 3
• A binary number 1111₂ = 1×2³ + 1×2² + 1×2¹ + 1×2⁰ = 8 + 4 + 2 + 1 = 15

Solution:

Set them equal:

2y + 3 = 15

2y = 12

y = 6

Check: 23 in base 6 = 2×6 + 3 = 15 ✓, and 1111₂ = 15 ✓. Also, the digit 3 is less than 6, so base 6 is valid.

Answer: y = 6, which is option C.

Tip: For base-conversion questions, always change everything to base 10 first, then form a simple equation. Remember that 1111₂ = 2⁴ − 1 = 15, a handy shortcut for any string of 1s in binary.