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If a and b are positive integers, is the sum a + b divisible by 4?
- When the sum 23a + 25b is divided by 10, the remainder is 8
- When 22b is divided by 10, the remainder is 8
- 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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Verified Answer
If a and b are positive integers, is the sum a + b divisible by 4? Whe...
Steps 1 & 2: Understand Question and Draw Inferences
Given: Integers a, b > 0
To find: If a + b is divisible by 4?
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘When the sum 23a + 25b is divided by 10, the remainder is 8’
- Now, we know that when an integer is divided by 10, the remainder is equal to the units digit of that integer
- For example, when 25 is divided by 10, the remainder is 5 (the units digit of 25)
- So, from Statement 1, we can infer that the units digit of the sum 23a + 25b is 8
- Now, the units digit of 25b will always be 5, for all possible values of the positive integer b
- Therefore, the units digit of the term 23a must be 3
- Now, the units digit of 23a comes from 3a
- For the units digit to be 3, a must be of the form 4k + 1, where k is a non-negative integer
- Thus, from Statement 1, we have inferred that a is of the form 4k + 1. However, since we do not yet know about the value of b, we cannot comment if the sum a + b is divisible by 4 or not.
- So, Statement 1 is not sufficient.
Step 4: Analyze Statement 2 independently
Statement 2 says that ‘When 22b is divided by 10, the remainder is 8’
- We know that when an integer is divided by 10, the remainder is equal to the units digit of that integer
- So, from Statement II, we can infer that the units digit of 22b is 8
- Now, the units digit of 22b comes from 2b
- For the units digit to be 8, b must be of the form 4m + 3, where m is a non-negative integer
- Thus, from Statement 2, we have inferred that b is of the form 4m + 3. However, since we do not know about the value of a, we cannot comment if the sum a + b is divisible by 4 or not.
- So, Statement 2 is not sufficient.
Step 5: Analyze Both Statements Together (if needed)
- From Statement 1, we inferred that a is of the form 4k + 1
- From Statement 2, we inferred that b is of the form 4m + 3
- So, combining the 2 statements, we observe that the sum a + b is of the form 4k + 4m + 4
- Therefore the sum a + b is indeed divisible by 4
- Since we’ve arrived at a unique answer to the question, the two statements together are sufficient.
Answer: Option C
Most Upvoted Answer
If a and b are positive integers, is the sum a + b divisible by 4? Whe...
Statement 1: When the sum a + b is divided by 4, the remainder is 2.
Statement 2: When 23a + 25b is divided by 10, the remainder is 8. When 22b is divided by 10, the remainder is 8.
To determine if the sum a + b is divisible by 4, we need to consider the possible remainders when a and b are divided by 4.
Analysis of Statement 1:
When a + b is divided by 4, the remainder is 2.
Possible combinations of remainders when dividing a and b by 4 that satisfy this condition are:
- a = 0 (divisible by 4) and b = 2 (remainder 2 when divided by 4)
- a = 2 (remainder 2 when divided by 4) and b = 0 (divisible by 4)
- a = 1 (remainder 1 when divided by 4) and b = 1 (remainder 1 when divided by 4)
In all these cases, the sum a + b is divisible by 4. Therefore, Statement 1 alone is sufficient to answer the question.
Analysis of Statement 2:
When 23a + 25b is divided by 10, the remainder is 8.
When 22b is divided by 10, the remainder is 8.
From these two statements, we can deduce the following:
- The remainder when 23a is divided by 10 is 0 (since the remainder when 23a + 25b is divided by 10 is 8, and the remainder when 25b is divided by 10 is 8).
- The remainder when 22b is divided by 10 is 8.
- This implies that b is not divisible by 5 (since the remainder when 22b is divided by 10 is 8, which means the remainder when b is divided by 5 is not 0).
However, we cannot determine the remainder when a + b is divided by 4 from this information alone. Therefore, Statement 2 alone is not sufficient to answer the question.
Conclusion:
From the analysis of both statements, we can conclude that:
- Statement 1 alone is sufficient to determine that the sum a + b is divisible by 4.
- Statement 2 alone is not sufficient to determine if the sum a + b is divisible by 4.
- Therefore, both statements together are sufficient to answer the question.
Statement 2: When 23a + 25b is divided by 10, the remainder is 8. When 22b is divided by 10, the remainder is 8.
To determine if the sum a + b is divisible by 4, we need to consider the possible remainders when a and b are divided by 4.
Analysis of Statement 1:
When a + b is divided by 4, the remainder is 2.
Possible combinations of remainders when dividing a and b by 4 that satisfy this condition are:
- a = 0 (divisible by 4) and b = 2 (remainder 2 when divided by 4)
- a = 2 (remainder 2 when divided by 4) and b = 0 (divisible by 4)
- a = 1 (remainder 1 when divided by 4) and b = 1 (remainder 1 when divided by 4)
In all these cases, the sum a + b is divisible by 4. Therefore, Statement 1 alone is sufficient to answer the question.
Analysis of Statement 2:
When 23a + 25b is divided by 10, the remainder is 8.
When 22b is divided by 10, the remainder is 8.
From these two statements, we can deduce the following:
- The remainder when 23a is divided by 10 is 0 (since the remainder when 23a + 25b is divided by 10 is 8, and the remainder when 25b is divided by 10 is 8).
- The remainder when 22b is divided by 10 is 8.
- This implies that b is not divisible by 5 (since the remainder when 22b is divided by 10 is 8, which means the remainder when b is divided by 5 is not 0).
However, we cannot determine the remainder when a + b is divided by 4 from this information alone. Therefore, Statement 2 alone is not sufficient to answer the question.
Conclusion:
From the analysis of both statements, we can conclude that:
- Statement 1 alone is sufficient to determine that the sum a + b is divisible by 4.
- Statement 2 alone is not sufficient to determine if the sum a + b is divisible by 4.
- Therefore, both statements together are sufficient to answer the question.
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If a and b are positive integers, is the sum a + b divisible by 4? When the sum 23a + 25b is divided by 10, the remainder is 8 When 22b is divided by 10, the remainder is 8a)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 a and b are positive integers, is the sum a + b divisible by 4? When the sum 23a + 25b is divided by 10, the remainder is 8 When 22b is divided by 10, the remainder is 8a)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 a and b are positive integers, is the sum a + b divisible by 4? When the sum 23a + 25b is divided by 10, the remainder is 8 When 22b is divided by 10, the remainder is 8a)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 a and b are positive integers, is the sum a + b divisible by 4? When the sum 23a + 25b is divided by 10, the remainder is 8 When 22b is divided by 10, the remainder is 8a)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?.
If a and b are positive integers, is the sum a + b divisible by 4? When the sum 23a + 25b is divided by 10, the remainder is 8 When 22b is divided by 10, the remainder is 8a)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 a and b are positive integers, is the sum a + b divisible by 4? When the sum 23a + 25b is divided by 10, the remainder is 8 When 22b is divided by 10, the remainder is 8a)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 a and b are positive integers, is the sum a + b divisible by 4? When the sum 23a + 25b is divided by 10, the remainder is 8 When 22b is divided by 10, the remainder is 8a)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?.
Solutions for If a and b are positive integers, is the sum a + b divisible by 4? When the sum 23a + 25b is divided by 10, the remainder is 8 When 22b is divided by 10, the remainder is 8a)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? in English & in Hindi are available as part of our courses for UPSC.
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If a and b are positive integers, is the sum a + b divisible by 4? When the sum 23a + 25b is divided by 10, the remainder is 8 When 22b is divided by 10, the remainder is 8a)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?, a detailed solution for If a and b are positive integers, is the sum a + b divisible by 4? When the sum 23a + 25b is divided by 10, the remainder is 8 When 22b is divided by 10, the remainder is 8a)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? has been provided alongside types of If a and b are positive integers, is the sum a + b divisible by 4? When the sum 23a + 25b is divided by 10, the remainder is 8 When 22b is divided by 10, the remainder is 8a)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? theory, EduRev gives you an
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