Solution:
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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Now we will solve this DS question using the Variable Approach.
Let’s apply the 3 steps suggested previously.
Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.
We have to find the values of natural numbers p, q, and r.
Follow the second and third steps: From the original condition, we have 3 variables (p, q, and r). To match the number of variables with the number of equations, we need 3 more equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.
Recall 3 Principles and choose E as the most likely answer
Let's take a look at both conditions (1) and (2) together. Then we get:
=> p < q < r and pqr = 5 * (p + q + r)
=> p = 2 , q = 5 and r = 7
=> pqr = 70 and p + q + r = 14
Since the answer is unique, both conditions (1) and (2) combined are sufficient, according to CMT 2, which states that the number of answers must be one. So, C seems to be the answer.
However, since this question is an integer question, which is also one of the key questions, we should apply CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer. Let’s look at each condition separately.
Condition (1) tells us that p, q, and r are prime numbers and p < q < r.
=> If p = 2, q = 3, and r = 5 then p < q < r.
=> If p = 3, q = 5, and r = 7 then p < q < r.
Since the answer is not unique, the condition is not sufficient, according to CMT 2, which states that the number of answers must be one.
Condition (2) tells us that the product of p, q, and r is 5 times the sum of p, q, and r.
=> Thus, pqr = 5 (p + q + r). This means that pqr is a multiple of 5, and thus one of the prime numbers is 5.
=> If p = 5 , q = 7, and r = 2 then pqr = 5 * 7 * 2 = 70 and p + q + r = 5 + 7 + 2 = 14. Therefore, pqr = 5 (p + q + r).
However, if p = 2, q = 7, and r = 5 then pqr = 2 * 7 * 5 = 70 and p + q + r = 2 + 7 + 5 = 14. Therefore, pqr = 5 (p + q + r).
Since the answer is not unique, the condition is not sufficient, according to CMT 2, which states that the number of answers must be one.
So, both conditions (1) and (2) together are sufficient.
Therefore, C is the correct answer.
Answer C
