The rule is simple and absolute: in base r, the only legal digits are 0 to r − 1. A number is valid in a base only if every one of its digits obeys that limit.

Test the digits of 132 — the largest is 3.

So 132 is invalid only in base 2 — option 4.

What the same string means in each valid base is worth seeing, because it makes the positional principle concrete:

\((132)_{8}=1\times8^{2}+3\times8^{1}+2\times8^{0}=64+24+2=90_{10}\)

\((132)_{16}=1\times256+3\times16+2=306_{10}\)

\((132)_{10}=132\)

The same three symbols denote three different quantities — which is why a base subscript is never optional when more than one base is in play.

The general rule that follows: the larger the base, the more values a given number of digits can represent, since n digits in base r span \(r^{n}\) values. Binary needs the most digits of all, which is exactly why hexadecimal is used as shorthand — one hex digit replaces four binary digits, with no arithmetic required for the conversion.

A practical note on validity checking : the quickest test is to find the largest digit present and add one; that is the smallest base in which the number is legal. Here the largest digit is 3, so 132 is valid in base 4 and every base above it, and invalid in bases 2 and 3.

Hence, 132 is not a valid number in base 2.