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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Test: Unit Digit- 2

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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...

Understanding the Problem
To solve the problem, we need to find the remainder when (x+y)a is divided by 10, given that x and y are distinct positive integers and x+y is even.
Key Observations
- Since x and y are distinct positive integers and x+y is even, both x and y must be either odd or even.
- The last digit of (x+y)a will determine the remainder when (x+y)a is divided by 10.
Analysis of Statements
Statement (1): Units digit of y is 6
- If the units digit of y is 6, then y can be represented as 6, 16, 26, etc.
- For x+y to be even, x must also be even (since an even number added to an even number is even).
- The last digit of an even number could be 0, 2, 4, 6, or 8.
- Thus, (x+y) will always have a last digit that is even (6, 8, 0, etc.), but we cannot determine a specific last digit.
- Without knowing a, we cannot determine the last digit of (x+y)a.
Statement (2): (xy)a is divisible by 10
- If (xy)a is divisible by 10, then either x or y must end with 0 or 5.
- If either x or y ends with 0, then x+y will also end in 0, hence (x+y)a will have a last digit of 0.
- If either ends with 5, then both must end with an even number (to satisfy x+y being even), making the last digit of (x+y)a even.
- However, without knowing a, we cannot determine the last digit conclusively.
Combining Statements
- From Statement (1), we know y ends with 6 and x must be even.
- From Statement (2), we know (xy)a is divisible by 10, meaning either x or y must contribute a factor of 10.
- Together, they ensure (x+y)a will yield a definite last digit.
Conclusion
- Both statements together provide enough information to conclude the last digit of (x+y)a.
- Therefore, the correct answer is option 'C': BOTH statements are sufficient together but neither is sufficient alone.

Answered by Notes Wala · View profile

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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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