If (y)x represents a number y in base x, then which of the following numbers…

Concept

In a positional numeral system with base b, the value of a four-digit numeral (d3d2d1d0)b is d3b3 + d2b2 + d1b + d0.

Therefore, (1111)b = b3 + b2 + b + 1. To compare numerals written in different bases, first express each in the same base, such as decimal.

Application

1. For base 2: (1111)2 = 1×23 + 1×22 + 1×2 + 1 = 8 + 4 + 2 + 1 = 15.

2. For base 8: (1111)8 = 1×83 + 1×82 + 1×8 + 1 = 512 + 64 + 8 + 1 = 585.

3. For base 10: (1111)10 = 1×103 + 1×102 + 1×10 + 1 = 1000 + 100 + 10 + 1 = 1111.

4. For base 16: (1111)16 = 1×163 + 1×162 + 1×16 + 1 = 4096 + 256 + 16 + 1 = 4369.

Cross-check

For b ≥ 2, the expression b3 + b2 + b + 1 increases as b increases. Since 2 is the smallest listed base, its numeral must have the smallest value; this agrees with the decimal conversions.

Result

The smallest number is (1111)2, whose decimal value is 15.