Valid base for given digits – UGC NET Question

In any positional number system of radix r, the permissible digits range from 0 to r−1. If a numeral includes the digit 3, then the base must be at least 4, because in base 3 the highest digit allowed is 2. Therefore, the smallest base that can legally contain the digits 0, 1, 2 and 3 is base 4.

Option A:
Option A recognises that the presence of digit 3 forces r−1 ≥ 3, so r ≥ 4. Among the given options, base 4 is the minimal base satisfying this inequality. Thus, it is the smallest valid radix for such numerals.

Option B:
Option B, base 3, can only use digits 0, 1, and 2, since 3 itself is not a valid digit in radix 3. Hence any numeral containing the digit 3 would be invalid in base 3, making this option incorrect.

Option C:
Option C, base 5, is a permissible base for digits 0–3, but it is not the smallest acceptable base. It satisfies r ≥ 4, yet the question specifically asks for the smallest radix, so base 4 is preferred.

Option D:
Option D, base 2, allows only the digits 0 and 1. It cannot represent numerals that contain the digits 2 or 3. Therefore, it is clearly too small to accommodate the given digit set.