Consider the given number (45)y where y is the base of the number. Some of the possible values of y are given below. (A) 5 (B) 6 (C) 7 (D) 8 Choose the correct answer from the options given below :
(B), (C) and (D) Only
In a positional number system with base y, the allowed digits are 0, 1, …, y − 1. A numeral is valid only if every digit is smaller than the base.
Step 1 — The largest digit in (45)y is 5, so we need \(y \gt 5\), i.e. \(y \ge 6\).
Step 2 — y = 5: digit 5 is not allowed in base 5 (digits 0–4), so (A) is not possible.
Step 3 — y = 6, 7, 8 are all valid: \((45)_6 = 4 \times 6 + 5 = 29\), \((45)_7 = 33\), \((45)_8 = 37\).
Every combination that includes base 5 is therefore wrong, and the one that drops base 6 omits a valid base.
(B), (C) and (D) Only option is correct.
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