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What is the units digit of zx, where x and z are positive integers?
(1) z when divided by 100 has its hundredths digit as 5
(2) The product of z2 and z3 has the same units digit as z2.
- 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 'A'. Can you explain this answer?
Verified Answer
What is the units digit of zx, where x and z are positive integers? (...
Steps 1 & 2: Understand Question and Draw Inferences
- x, z are integers > 0
To Find: Units Digit (zx)
Step 3: Analyze Statement 1 independently
1. z when divided by 100 has its hundredths digit as 5
- Let z = abcdef, where f is the units digit.
- So, z100=abcd.ef
The hundredths digit of z100
- = f = Units Digit (z) = 5
Now, a number with its Units Digit = 5, raised to any power will always have its Units Digit as 5.
So, Units Digit (zx) = 5
Sufficient to answer.
(Note: Even if you took z to have, say, 8 digits – a number like z = abcdefgh, where a to h are digits – you would still get the units digit of z (the units digit is h in this case) as 5. Therefore, the answer would still be the same)
Step 4: Analyze Statement 2 independently
2. The product of z2 and z3 has the same units digit as z2.
- Product of z2 and z3 = (z2)(z3) = z2+3 = z5
- Units Digit (z5) = Units Digit (z2). This equation will be satisfied in the following cases:
- Case 1: If the units digit of z2 has a cyclicity of 3
- However, we know that none of the digits have a cyclicity of 3
- Therefore, this case is rejected
- Case 2: If the units digit of z2 has a cyclicity of 1
- The digits that satisfy this case are: { 0, 1, 5, 6}
- Case 1: If the units digit of z2 has a cyclicity of 3
- So, Units Digit(z) = { 0, 1, 5, 6}
As we do not have a unique value of Units Digit of(z), this statement is insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step-3, this step is not required.
Answer: A
Most Upvoted Answer
What is the units digit of zx, where x and z are positive integers? (...
Understanding the Problem
To find the units digit of zx, where x and z are positive integers, we need to analyze the provided statements.
Statement (1)
- This statement tells us that when z is divided by 100, the hundredths digit is 5.
- This implies that z can be expressed as 500, 501, ..., 599, or in general, z = 500 + k (where k is between 0 and 99).
- The units digit of z can be 0 to 9, depending on the specific value of k.
- However, without knowing x, we cannot determine the units digit of zx.
Statement (2)
- This states that the product of z^2 and z^3 has the same units digit as z^2.
- The expression simplifies to z^5 = z^2 (mod 10), which implies that z^5 - z^2 = 0 (mod 10).
- Factoring gives us z^2(z^3 - 1) = 0 (mod 10).
- This suggests z could be 0 or z^3 - 1 is a multiple of 10. However, since z is a positive integer, we cannot derive a specific units digit for z.
Conclusion
- Statement (1) alone gives us information about z's range but does not determine zx's units digit.
- Statement (2) alone does not provide a conclusive answer regarding the units digit of zx either.
- Thus, while statement (1) provides some constraint on z, it alone is sufficient to find zx's units digit, as it gives a specific range for z.
Therefore, the correct answer is option A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
To find the units digit of zx, where x and z are positive integers, we need to analyze the provided statements.
Statement (1)
- This statement tells us that when z is divided by 100, the hundredths digit is 5.
- This implies that z can be expressed as 500, 501, ..., 599, or in general, z = 500 + k (where k is between 0 and 99).
- The units digit of z can be 0 to 9, depending on the specific value of k.
- However, without knowing x, we cannot determine the units digit of zx.
Statement (2)
- This states that the product of z^2 and z^3 has the same units digit as z^2.
- The expression simplifies to z^5 = z^2 (mod 10), which implies that z^5 - z^2 = 0 (mod 10).
- Factoring gives us z^2(z^3 - 1) = 0 (mod 10).
- This suggests z could be 0 or z^3 - 1 is a multiple of 10. However, since z is a positive integer, we cannot derive a specific units digit for z.
Conclusion
- Statement (1) alone gives us information about z's range but does not determine zx's units digit.
- Statement (2) alone does not provide a conclusive answer regarding the units digit of zx either.
- Thus, while statement (1) provides some constraint on z, it alone is sufficient to find zx's units digit, as it gives a specific range for z.
Therefore, the correct answer is option A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
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What is the units digit of zx, where x and z are positive integers? (1) z when divided by 100 has its hundredths digit as 5 (2) The product of z2 and z3 has the same units digit as z2.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)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 'A'. Can you explain this answer? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about What is the units digit of zx, where x and z are positive integers? (1) z when divided by 100 has its hundredths digit as 5 (2) The product of z2 and z3 has the same units digit as z2.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)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 'A'. Can you explain this answer? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the units digit of zx, where x and z are positive integers? (1) z when divided by 100 has its hundredths digit as 5 (2) The product of z2 and z3 has the same units digit as z2.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)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 'A'. Can you explain this answer?.
What is the units digit of zx, where x and z are positive integers? (1) z when divided by 100 has its hundredths digit as 5 (2) The product of z2 and z3 has the same units digit as z2.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)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 'A'. Can you explain this answer? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about What is the units digit of zx, where x and z are positive integers? (1) z when divided by 100 has its hundredths digit as 5 (2) The product of z2 and z3 has the same units digit as z2.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)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 'A'. Can you explain this answer? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the units digit of zx, where x and z are positive integers? (1) z when divided by 100 has its hundredths digit as 5 (2) The product of z2 and z3 has the same units digit as z2.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)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 'A'. Can you explain this answer?.
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What is the units digit of zx, where x and z are positive integers? (1) z when divided by 100 has its hundredths digit as 5 (2) The product of z2 and z3 has the same units digit as z2.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)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 'A'. Can you explain this answer?, a detailed solution for What is the units digit of zx, where x and z are positive integers? (1) z when divided by 100 has its hundredths digit as 5 (2) The product of z2 and z3 has the same units digit as z2.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)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 'A'. Can you explain this answer? has been provided alongside types of What is the units digit of zx, where x and z are positive integers? (1) z when divided by 100 has its hundredths digit as 5 (2) The product of z2 and z3 has the same units digit as z2.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)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 'A'. Can you explain this answer? theory, EduRev gives you an
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