If x and y are distinct positive integers and x+y is even, what is the remainder when (x+y)a is divided by 10, where a is a positive integer? (1) Units digit of y is 6 (2) (xy)a is divisible by 10.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'C'. Can you explain this answer?

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If x and y are distinct positive integers and x+y is even, what is the remainder when (x+y)^a is divided by 10, where a is a positive integer?

(1) Units digit of y is 6

(2) (xy)^a is divisible by 10.

a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Correct answer is option 'C'. Can you explain this answer?

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If x and y are distinct positive integers and x+y is even, what is the...

Steps 1 & 2: Understand Question and Draw Inferences

x, y are distinct integers > 0 such that x + y = even
- This means, Either both x andy are even
- Or both x andy are odd
a is an integer > 0

To Find: the value of r in (x+y)^a=10k+r, where k is the quotient obtained when (x+y)^a is divided by 10 and r is the remainder; so, 0 ≤ r < 10

Now, when a number is divided by 10, the remainder is equal to the units digit of that number.
So, r= units digit of (x+y)^a

Step 3: Analyze Statement 1 independently

1. Units digit of y is 6

It does not tell us anything about the units digit of x as well as about the value of a.

Hence, insufficient to answer.

Step 4: Analyze Statement 2 independently

2. (xy)^a is divisible by 10.

As (xy)^a is divisible by 10, the units digit of (xy)^a = 0
So, the units digit of xy = 0. Two cases are possible:
- Units digit of (x, y) = { 5, even number) in any order. However in this case the number with 5 as its units digit will be odd and the other number will be even. However, we’ve deduced in Steps 1 and 2 that x and y have the same even-odd nature. So, this case is not possible as it contradicts the given information (that the sum x + y is even).
- Units digit of (x, y) = (0, even number) in any order. In this case x and y are both even. So, this case is possible.
  - However since we do not have a unique value of units digit of both x and y and we do not know the value of a, we cannot find a unique value of the units digit of (x+y)^a

Insufficient to answer.

Step 5: Analyze Both Statements Together (if needed)

From Statement 1, we know that Units digit of y = 6
From Statement 2, we inferred that Units digit of (x, y) = (0, even number) in any order

Combining both the statements, we can say that units digit (x) = 0 and units digit(y) = 6

So, units digit of (x+y) = 6. Now do we need the value of a to find out the units digit of (x+y)?

We know a number with units digit of 6 raised to any power always results in units digit of 6.

So, Units Digit ((6^a)) = 6.

Thus r = Units Digit ((6^a)) = 6.

Sufficient to answer.

Answer: C

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If x and y are distinct positive integers and x+y is even, what is the remainder when (x+y)a is divided by 10, where a is a positive integer? (1) Units digit of y is 6 (2) (xy)a is divisible by 10.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'C'. Can you explain this answer?

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If x and y are distinct positive integers and x+y is even, what is the remainder when (x+y)a is divided by 10, where a is a positive integer? (1) Units digit of y is 6 (2) (xy)a is divisible by 10.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'C'. Can you explain this answer? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If x and y are distinct positive integers and x+y is even, what is the remainder when (x+y)a is divided by 10, where a is a positive integer? (1) Units digit of y is 6 (2) (xy)a is divisible by 10.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If x and y are distinct positive integers and x+y is even, what is the remainder when (x+y)a is divided by 10, where a is a positive integer? (1) Units digit of y is 6 (2) (xy)a is divisible by 10.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'C'. Can you explain this answer?.

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