In-Depth: Remainders (Remainder Theorem & Cyclicity)
CLAT Application & Relevance
Importance: Low. Remainders, especially complex ones involving cyclicity or advanced theorems, are highly unlikely to be directly tested in CLAT QT. However, a basic understanding of remainders is part of fundamental number sense. Simple remainder-based questions might appear as logical deductions within a very basic numerical reasoning passage, but focus on the underlying concept rather than advanced theorems.
How it's tested: Very rarely, as simple direct questions involving division, or implicitly when dealing with properties of numbers in a broader numerical reasoning context.
Section 1: Core Concepts & Methods
When an integer is divided by another integer, the result is often a quotient and a remainder. The remainder is the amount "left over" after division.
Basic Remainder Concept
For any integers 'a' (dividend) and 'b' (divisor), where b ≠ 0:
a = bq + r
Where:
- is the quotient
- is the remainder, such that
0 ≤ r < |b|
Example: When 17 is divided by 5: 17 = 5 × 3 + 2. Here, 2 is the remainder.
Key Concepts & Methods
- Remainder Theorem (for Polynomials): If a polynomial
P(x)is divided by(x-a), the remainder isP(a). While important in higher algebra, its direct application is extremely unlikely in CLAT QT which focuses on numerical aptitude. - Cyclicity of Remainders: For powers of a number divided by another number, the remainders often follow a repeating pattern (cycle).
Example: Remainders of powers of 2 when divided by 3:- 2¹ ÷ 3 -> Remainder 2
- 2² ÷ 3 -> 4 ÷ 3 -> Remainder 1
- 2³ ÷ 3 -> 8 ÷ 3 -> Remainder 2
- 2⁴ ÷ 3 -> 16 ÷ 3 -> Remainder 1
- Negative Remainders: Sometimes, it's easier to use a negative remainder in intermediate steps. If a number divided by 'd' gives a remainder of -r, the actual remainder is
d - r. (e.g., 17 ÷ 5, can be thought of as 3x5=15, rem 2. Or 4x5=20, rem -3. Actual rem = 5-3=2).
Section 2: Solved CLAT-Style Examples (Basic Application)
Example 1: Basic Remainder Calculation (Caselet)
Passage Context: "A law firm assigns specific document codes. For a particular audit, all document codes are divided by 7, and the remainder is used to categorize the document's priority. One document has the code 125."
Question: "What is the remainder when 125 is divided by 7?"
Detailed Solution:
1. Perform Division:
Divide 125 by 7.
125 = 7 × 17 + 6
(7 * 10 = 70. Remaining = 55. 7 * 7 = 49. Remaining = 6. So 17 is quotient, 6 is remainder.)
Answer: The remainder is 6.
Example 2: Remainder in Practical Scenario (Caselet)
Passage Context: "A legal research server performs batch processing. It processes exactly 15 tasks per minute. If a new queue of 257 tasks arrives."
Question: "After how many full minutes of processing will there be tasks remaining, and how many tasks will be left over?"
Detailed Solution:
1. Divide Total Tasks by Tasks per Minute:
Divide 257 by 15.
257 ÷ 15
257 = 15 × 17 + 2
2. Interpret Quotient and Remainder:
Quotient = 17 (This means 17 full minutes of processing will occur).
Remainder = 2 (These are the tasks left over after 17 full minutes).
Answer: After 17 full minutes of processing, 2 tasks will be left over.