Forget the conventional way to solve DS questions.
We will solve this DS question using the variable approach.Remember the relation between the Variable Approach, and Common Mistake Types 3 and 4 (A and B)[Watch lessons on our website to master these approaches and tips]
Step 1: Apply Variable Approach(VA)Step II: After applying VA, if C is the answer, check whether the question is the key question.StepIII: If the question is not a key question, choose C as the probable answer, but if the question is a key question, apply CMT 3 and 4 (A or B).Step IV: If CMT3 or 4 (A or B) is applied, choose either A, B, or D.Let's apply CMT (2), which says there should be only one answer for the condition to be sufficient. Also, this is an integer question and, therefore, we will have to apply CMT 3 and 4 (A or B).
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Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]
Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.We have to find the units’ digit of \(7^x + 9^y\) - where 'x' and 'y' are positive integers and the product of x and y is prime.
Second and the third step of Variable Approach: From the original condition, we have 2 variables (x and y). To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 1 equation each, C would most likely be the answer.But we know that this is a key question [Integer question] and if we get an easy C as an answer, we will choose A or B.
Let’s take a look at each condition.Condition(1) tells us that 24 < y < 32.=> If the product of xy is a prime number then one of them has to be '1' and the other has to be a prime number. Since 'y' is between 24 and 32 hence x = 1.
=> For 'y' to be prime between 24 and 32, there are two prime numbers: 29 and 31.
=> For y = 29 = 9^{29} => 9^{odd power} = 9
=> For y = 31 = 9^{31} => 9^{odd power} = 9
=> For x = 1 = 7^{1} = 7
=> 7 + 9 = 6(unit digit)
Since the answer is a unique value, the condition (1) alone is sufficient by CMT 2.Condition(2) tells us that x = 1.=> 'Y' can be 2(even prime number) or other any prime numbers(all odd)
=> For y = 2 = 9^{2} => 9^{even power} = 1
=> For x = 1 = 7^{1} = 7
=> 7 + 1 = 8(unit digit)
=> For y = 31 = 9^{31} => 9^{odd power} = 9
=> For x = 1 = 7^{1} = 7
=> 7 + 9 = 6(unit digit)
Since the answer is not a unique value, the condition(2) is not sufficient by CMT 2. Condition (1) alone is sufficient.So, A is the correct answer.Answer: A