If x and y are positive integers such that the product of x

I do not agree with the OA. Answer should be E.

From the question statement, we can see that one of the numbers is 1 and the other is a prime.

1)We get x=1, y can be 29 or 31. If it is 29, units digit of expression is 8. If it is 31, units digit of expression is 6. Insufficient.

2) Obviously insufficient. As illustrated from statement 1.

Answer is hence E.

Kudos Please... If my post helped.

Since x*y is prime one of them must be 1 and other must be prime.

Stmt1: 28<y<32. Therefore x has to be 1. and y must be 29 or 31. Now 9 raised to odd power always has 9 in units digit. What is this final units digit? 7+9 = 6. SUFF.
Stmt2: x=1. y could be any prime under the sun. NOT SUFF.

First things first. Xy is a prime number can only be true when either x or y is a prime number and the other is equal yo 1. Now, Statement 1 tells us that y is a prime number in that range thus y (29,31) and x of course 1. Now When 9 is raised to an odd number the units digit is always 9 therefore Statement 1 is sufficient.

Statement 2 tells us that x=1, but Y could be any prime number so it is insufficient.

Therefore A is the correct answer

Cheers
J

First, remember that a prime has only 2 factors: 1 and itself. Thus, if x * y = prime, either x or y is 1 and the other is prime.
Now, look at the cyclicity of 7 and 9. 7^1 = 7, 7² = 49, 7³ = 343, 7^4 = 2401 --> next will be units digit of 7 again. Cyclicity is 4.
For 9 the rule is : If 9 is raised to an odd power, the units digit will be 9, if its raised to an even power the units digit will be 1.

(1) 24 < y < 32 --> y is not 1, hence X has to be 1. Units digit of 7^1 = 7. Now the only two primes in the given interval are 29 and 31, both of which are odd. Hence units digit of 9^y will be 9. Hence units digit of the sum is 0. SUFF.

(2) x = 1. This tells us again that 7^1 = 7 BUT y can now be 2 (which is the only EVEN prime) OR any other ODD prime. Thus the units digit of 9^y can either be 1 or 9. Insufficient.

Hope its all clear!

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).

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

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

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