Which of the following is the 4's complement of (1011)₅, that is in base 5…

To find the r's complement of a number in base r, we first calculate the (r-1)'s complement and then add 1 to it. Here, we need the 4's complement of (1011)₅ in base 5. First, determine the number of digits: there are 4 digits. The (r-1)'s complement in base r is found by subtracting each digit from (r-1). Since the base is 5, we subtract each digit of (1011)₅ from 4. Subtracting the first digit: 4 - 1 = 3. Second digit: 4 - 0 = 4. Third digit: 4 - 1 = 3. Fourth digit: 4 - 1 = 3. This gives the 4's complement as (3433)₅ directly because in base r, the r's complement is often defined as (r^n - N), but for digit-wise operations in base 5, the complement of a number with n digits is (r^n - N). However, strictly speaking, the 4's complement usually refers to the diminished radix complement (base-1) in some contexts, but here it implies finding the value such that adding it to the original yields 10000₅ (which is 5^4). Let's re-evaluate: The standard definition for r's complement of an n-digit number N in base r is (r^n - N). For 4's complement of a base-5 number, this phrasing is slightly ambiguous. Usually, we calculate the (r-1)'s complement first. If '4's complement' means the complement with respect to base 5 (which is r=5), then it's the 5's complement. If it literally means base 4, that doesn't make sense for a base-5 number. Assuming the question asks for the 5's complement (often confused or typoed as r's where r is base), let's calculate 5's complement: (10000)₅ - (1011)₅. 5-1=4, borrow... actually simpler: (r^n - N). 5^4 = 10000₅. 10000₅ - 1011₅ = (4-1)(4-0)(4-1)(5-1) -> 3, 4, 3, 4? No. Let's do subtraction: 10000 - 1011. Last digit: 5-1=4 (borrow). Next: 4-1=3. Next: 4-0=4. First: 4-1=3. Result (3434)₅? Wait, the correct answer is B: (3433)₅. This implies the question asks for the 4's complement where '4' is (r-1). So it is asking for the diminished radix complement (base 5, r-1 = 4). The rule is: subtract each digit from (r-1) = 4. Digit 1: 4-1=3. Digit 2: 4-0=4. Digit 3: 4-1=3. Digit 4: 4-1=3. Result is (3433)₅. This matches Option B. Options A, C, and D are incorrect because they do not follow the digit-wise subtraction from 4 rule. Option A is too small, C and D contain digits invalid for base 5 (8, 9) or incorrect values.