Divisibility and remainder concepts – UGC NET Question

Statements A, C and D summarise key properties of divisibility and remainders. A encodes the division algorithm representation. C is true because 6 has prime factors 2 and 3, so divisibility by 6 implies divisibility by both. D is correct since numbers with the same remainder differ by a multiple of the divisor. B is false as divisibility by 3 and 9 together does not force divisibility by 27, and E is false because a multiple of 4 need not be a multiple of 8. Therefore, A, C and D only are correct.

Option A:
Option A is correct because it collects the basic representation of numbers with remainders and two important consequences for divisibility, while excluding B and E, which overstate divisibility conditions.

Option B:
Option B is incorrect as it adds B, which wrongly claims that common divisibility by 3 and 9 forces divisibility by 27, contradicting simple counterexamples like 9 itself.

Option C:
Option C is wrong since it also incorporates E, which incorrectly asserts that multiples of 4 are automatically multiples of 8, and thus mixes false statements with true ones.

Option D:
Option D is incomplete because it drops A and only keeps C and D, leaving out the fundamental division algorithm expression that underpins the remainder concept in many NET questions.