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If z is a positive integer and r is the remainder when z2 + 2z + 4 is divided by 8, what is the value of r?
(1) When (z-3)2 is divided by 8, the remainder is 4
(2) When 2z is divided by 8, the remainder is 2
- 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 'D'. Can you explain this answer?
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Verified Answer
If z is a positive integer and r is the remainder when z2 + 2z + 4 is ...
Steps 1 & 2: Understand Question and Draw Inferences
Given:
- Positive integer z
- z2+2z+4=8m+r,
- where m is an integer and r is the remainder
To find: The value of r
Step 3: Analyze Statement 1 independently
- When (z-3)2 is divided by 8, the remainder is 4
- So, we can write: (z−3)2=8k+4 , where k is an integer
- That is, z2–6z+9=8k+4
- Let us try to re-express the above equation in terms of the expression in the question:
- ? z2+2z+4
- z2 – 6z +( 4+5) = 8k + 4
- So, z2–6z+4=8k−1
- Adding 8z to both sides:
- z2 + 2z + 4 = 8 (k + z) - 1
- Now the expression on the left hand side of the equation is exactly the same as the expression in the question. On the right hand side, we see that the term 8(k+z) will be divisible by 8. But the remainder cannot be -1 because remainder cannot be negative. So, let’s rewrite the above equation:
- z2 + 2z + 4 = 8(k + z) - 8 + 7
- z2 + 2z + 4 = 8(k + z - 1) + 7
- The equation now is exactly comparable to the form 8m + r
- So, the remainder r = 7
Thus, Statement 1 alone is sufficient to answer the question.
Step 4: Analyze Statement 2 independently
- When 2z is divided by 8, the remainder is 2
- So we can write, 2z = 8n + 2, where n is some integer
- That is, z = 4n + 1
- In the above equation, the term 16n(n+1)
- will be divisible by 8
- So, the remainder when z2+2z+4 is divided by 8 will be 7
Thus, Statement 2 alone is sufficient to find a unique value of the remainder
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in each of Steps 3 and 4, this step is not required
Answer: Option D
Most Upvoted Answer
If z is a positive integer and r is the remainder when z2 + 2z + 4 is ...
Understanding the Problem
To find the remainder r when z^2 + 2z + 4 is divided by 8, we can analyze the given statements individually and together.
Statement (1): When (z-3)² is divided by 8, the remainder is 4
- This means (z-3)² ≡ 4 (mod 8).
- Expanding gives z² - 6z + 9 ≡ 4 (mod 8).
- This simplifies to z² - 6z + 5 ≡ 0 (mod 8).
- Rearranging gives z² + 2z + 4 ≡ 0 (mod 8), hence r = 0.
Statement (2): When 2z is divided by 8, the remainder is 2
- This tells us 2z ≡ 2 (mod 8), implying z ≡ 1 (mod 4).
- Possible values for z: 1, 5, 9, etc.
- Calculate for z = 1: r = 1 + 2 + 4 = 7.
- Calculate for z = 5: r = 25 + 10 + 4 = 39, and 39 mod 8 = 7.
- So, r = 7 regardless of z being 1 or 5.
Combining Both Statements
- From statement (1), we know r = 0.
- From statement (2), we know r = 7.
- Each statement gives a consistent value for r independently.
Conclusion
- Both statements independently provide sufficient information to determine the value of r.
- Therefore, the correct answer is option 'D': Each statement alone is sufficient to answer the question.
To find the remainder r when z^2 + 2z + 4 is divided by 8, we can analyze the given statements individually and together.
Statement (1): When (z-3)² is divided by 8, the remainder is 4
- This means (z-3)² ≡ 4 (mod 8).
- Expanding gives z² - 6z + 9 ≡ 4 (mod 8).
- This simplifies to z² - 6z + 5 ≡ 0 (mod 8).
- Rearranging gives z² + 2z + 4 ≡ 0 (mod 8), hence r = 0.
Statement (2): When 2z is divided by 8, the remainder is 2
- This tells us 2z ≡ 2 (mod 8), implying z ≡ 1 (mod 4).
- Possible values for z: 1, 5, 9, etc.
- Calculate for z = 1: r = 1 + 2 + 4 = 7.
- Calculate for z = 5: r = 25 + 10 + 4 = 39, and 39 mod 8 = 7.
- So, r = 7 regardless of z being 1 or 5.
Combining Both Statements
- From statement (1), we know r = 0.
- From statement (2), we know r = 7.
- Each statement gives a consistent value for r independently.
Conclusion
- Both statements independently provide sufficient information to determine the value of r.
- Therefore, the correct answer is option 'D': Each statement alone is sufficient to answer the question.
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If z is a positive integer and r is the remainder when z2 + 2z + 4 is divided by 8, what is the value of r?(1)When (z-3)2 is divided by 8, the remainder is 4(2) When 2z is divided by 8, the remainder is 2a)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 'D'. Can you explain this answer?
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If z is a positive integer and r is the remainder when z2 + 2z + 4 is divided by 8, what is the value of r?(1)When (z-3)2 is divided by 8, the remainder is 4(2) When 2z is divided by 8, the remainder is 2a)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 'D'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If z is a positive integer and r is the remainder when z2 + 2z + 4 is divided by 8, what is the value of r?(1)When (z-3)2 is divided by 8, the remainder is 4(2) When 2z is divided by 8, the remainder is 2a)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 'D'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If z is a positive integer and r is the remainder when z2 + 2z + 4 is divided by 8, what is the value of r?(1)When (z-3)2 is divided by 8, the remainder is 4(2) When 2z is divided by 8, the remainder is 2a)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 'D'. Can you explain this answer?.
If z is a positive integer and r is the remainder when z2 + 2z + 4 is divided by 8, what is the value of r?(1)When (z-3)2 is divided by 8, the remainder is 4(2) When 2z is divided by 8, the remainder is 2a)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 'D'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If z is a positive integer and r is the remainder when z2 + 2z + 4 is divided by 8, what is the value of r?(1)When (z-3)2 is divided by 8, the remainder is 4(2) When 2z is divided by 8, the remainder is 2a)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 'D'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If z is a positive integer and r is the remainder when z2 + 2z + 4 is divided by 8, what is the value of r?(1)When (z-3)2 is divided by 8, the remainder is 4(2) When 2z is divided by 8, the remainder is 2a)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 'D'. Can you explain this answer?.
Solutions for If z is a positive integer and r is the remainder when z2 + 2z + 4 is divided by 8, what is the value of r?(1)When (z-3)2 is divided by 8, the remainder is 4(2) When 2z is divided by 8, the remainder is 2a)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 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for GMAT.
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ample number of questions to practice If z is a positive integer and r is the remainder when z2 + 2z + 4 is divided by 8, what is the value of r?(1)When (z-3)2 is divided by 8, the remainder is 4(2) When 2z is divided by 8, the remainder is 2a)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 'D'. Can you explain this answer? tests, examples and also practice GMAT tests.
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