A certain number gives a quotient of 271 and a remainder of 0 when divided by 96. If the same number is divided by 95, what remainder will be obtained?

Introduction / Context:
Remainder and quotient problems like this one are fundamental in number system questions. They test understanding of the division algorithm and how to manipulate a number when the divisor changes. Instead of working with the full division every time, we can use algebraic representations of the number in terms of divisor, quotient, and remainder. Here, we first reconstruct the number from one division and then redivide it by a different divisor to find the new remainder.

Given Data / Assumptions:

• When the number N is divided by 96, quotient is 271 and remainder is 0.
• This means N is exactly divisible by 96.
• We then divide the same N by 95.
• We must find the remainder when N is divided by 95.

Concept / Approach:
The division algorithm states that for a divisor d, quotient q and remainder r, the number N can be written as N = d * q + r. Since the remainder is 0 in the first division, N is exactly equal to 96 * 271. Once we compute N, we apply the division algorithm again with divisor 95 to determine the remainder. Instead of performing long division manually, we can use simple multiplication and modular reasoning to find the remainder quickly.

Step-by-Step Solution:
Step 1: Let the number be N. Step 2: From the given information, N = 96 * 271 + 0, so N = 96 * 271. Step 3: Compute N: 96 * 271 = 26016. Step 4: Now divide N by 95 and find the remainder. Step 5: Use the division idea: N = 95 * q + r, where r is the remainder and 0 ≤ r < 95. Step 6: Compute 26016 / 95. We find the integer part of the quotient first. Step 7: 95 * 273 = 25935 and 95 * 274 = 26030. Step 8: Since 26030 is greater than 26016, the quotient is 273. Step 9: Compute the remainder: r = 26016 − 95 * 273 = 26016 − 25935 = 81. Step 10: Hence, the remainder when N is divided by 95 is 81.

Verification / Alternative check:
As an alternative, we can note that 96 is 1 more than 95. Therefore N = 96 * 271 can be written as (95 + 1) * 271 = 95 * 271 + 271. When divided by 95, 95 * 271 gives remainder 0, and 271 gives remainder 271 mod 95. Since 95 * 2 = 190 and 95 * 3 = 285, we have 271 − 190 = 81. This again shows that the remainder is 81, confirming our previous calculation.

Why Other Options Are Wrong:
Option (a) 87: Does not match the remainder derived from either direct division or modular reasoning.
Option (b) 59: Also inconsistent with the remainder calculation of 81.
Option (c) 92: Greater than the correct remainder and does not satisfy the division relationship for N.

Common Pitfalls:
Some learners mistakenly try to divide 271 by 95 instead of reconstructing N. Others forget that the number is exactly divisible by 96 and might mis-handle the remainder. Arithmetic mistakes in multiplication or subtraction steps can also lead to wrong remainders. Working systematically and, where possible, using the representation N = (95 + 1) * 271 helps reduce errors.

Final Answer:
The remainder when the number is divided by 95 is 81.