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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember that equal number of variables and independent equations ensures a solution.

We have 1 variable and 0 equation from the original condition. Therefore, D is most likely to be the answer.

Condition 1)

Since n is odd, n-1 and n+1 are consecutive even integers.

One of two consecutive even integers must be a multiple of 4. For example, (2,4), (6,8), (8,10) and (10,12).

Thus (n-1)(n+1) is a multiple of 4, but its divisibility by 3 is not identified.

Counterexamples are n = 3 and n = 5.

(3-1)(3+1) = 2*4 = 8, which is not a multiple of 24.

(5-1)(5+1) = 4*6 = 24, which is a multiple of 24.

This is not sufficient.

Condition 2)

Since n is not a multiple of 3, n = 3k +1 or n = 3k + 2 for some integer k.

For the case, n = 3k +1, (n-1)(n+1) = (3k+1-1)(3k+1+1) = 3k(3k+2) is a multiple of 3. However, we can’t identify if n is a multiple of 24. Counterexamples are n = 4 for which (n-1)(n+1) = 3*5 = 15 is not a multiple of 24 and n = 7 for which (n-1)(n+1) = 6*8 = 48 is a multiple of 24.

For the case, n = 3k +2, (n-1)(n+1) = (3k+2-1)(3k+2+1) = (3k+1)(3k+3) = 3(3k+1)(k+1) is a multiple of 3. However, we can’t identify if n is a multiple of 24. Counterexamples are n = 5 for which (n-1)(n+1) = 4*6 = 15 is a multiple of 24 and n = 6 for which (n-1)(n+1) = 5*7 = 35 is a not multiple of 24.

Condition 1) & 2)

From the condition 1), (n-1)(n+1) is a multiple of 8 and from the condition 2), (n-1)(n+1) is a multiple of 3.

Thus, (n-1)(n+1) is a multiple of 24.

Therefore, unlike our expectation, C is the answer.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both con 1) and con 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using con 1) and con 2) separately according to DS definition. Obviously, there may be cases where the answer is A, B, C or E.

Answer: C

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