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If x and y are positive integers, is the sum x + y divisible by 4 ?

(1) When the sum 23x + 25y is divided by 10, the remainder is 8.
(2) When 22y 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
  • 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 are needed
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
If x and y are positive integers, is the sum x + y divisible by 4 ?(1)...
Statement (1): When the sum 23x + 25y is divided by 10, the remainder is 8.
We can rewrite this statement as (23x + 25y) ≡ 8 (mod 10). Simplifying further, we get (3x + 5y) ≡ 8 (mod 10).
From this information alone, we cannot directly determine if x + y is divisible by 4. For example, if x = 1 and y = 3, then (3x + 5y) ≡ (3 + 15) ≡ 8 (mod 10), but x + y is not divisible by 4. However, if x + y is divisible by 4, then (3x + 5y) ≡ 8 (mod 10) would hold true. Therefore, statement (1) alone is not sufficient.
Statement (2): When 22y is divided by 10, the remainder is 8.
We can rewrite this statement as 22y ≡ 8 (mod 10). Simplifying further, we get 2y ≡ 8 (mod 10).
Similar to statement (1), we cannot determine if x + y is divisible by 4 based solely on this information. For example, if y = 4, then 2y ≡ 8 (mod 10), but we don't know the value of x or if x + y is divisible by 4. Therefore, statement (2) alone is not sufficient.
However, by combining both statements, we have the system of congruences:
(3x + 5y) ≡ 8 (mod 10)
2y ≡ 8 (mod 10)
From the second congruence, we can deduce that y ≡ 4 (mod 10). This means that y must end with the digit 4. We can then substitute this result into the first congruence to obtain:
(3x + 5(10a + 4)) ≡ 8 (mod 10)
3x + 50a + 20 ≡ 8 (mod 10)
3x ≡ 8 - 20 (mod 10)
3x ≡ -12 (mod 10)
3x ≡ 8 (mod 10)
This congruence tells us that 3x ends with the digit 8. The only possible positive integer value for x that satisfies this condition is x = 6. When x = 6, the sum x + y is 6 + (10a + 4), which simplifies to 10a + 10, and is clearly divisible by 4.
Therefore, by considering both statements together, we can determine that the sum x + y is divisible by 4. Thus, the answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
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Most Upvoted Answer
If x and y are positive integers, is the sum x + y divisible by 4 ?(1)...
Statement (1): When the sum 23x + 25y is divided by 10, the remainder is 8.
We can rewrite this statement as (23x + 25y) ≡ 8 (mod 10). Simplifying further, we get (3x + 5y) ≡ 8 (mod 10).
From this information alone, we cannot directly determine if x + y is divisible by 4. For example, if x = 1 and y = 3, then (3x + 5y) ≡ (3 + 15) ≡ 8 (mod 10), but x + y is not divisible by 4. However, if x + y is divisible by 4, then (3x + 5y) ≡ 8 (mod 10) would hold true. Therefore, statement (1) alone is not sufficient.
Statement (2): When 22y is divided by 10, the remainder is 8.
We can rewrite this statement as 22y ≡ 8 (mod 10). Simplifying further, we get 2y ≡ 8 (mod 10).
Similar to statement (1), we cannot determine if x + y is divisible by 4 based solely on this information. For example, if y = 4, then 2y ≡ 8 (mod 10), but we don't know the value of x or if x + y is divisible by 4. Therefore, statement (2) alone is not sufficient.
However, by combining both statements, we have the system of congruences:
(3x + 5y) ≡ 8 (mod 10)
2y ≡ 8 (mod 10)
From the second congruence, we can deduce that y ≡ 4 (mod 10). This means that y must end with the digit 4. We can then substitute this result into the first congruence to obtain:
(3x + 5(10a + 4)) ≡ 8 (mod 10)
3x + 50a + 20 ≡ 8 (mod 10)
3x ≡ 8 - 20 (mod 10)
3x ≡ -12 (mod 10)
3x ≡ 8 (mod 10)
This congruence tells us that 3x ends with the digit 8. The only possible positive integer value for x that satisfies this condition is x = 6. When x = 6, the sum x + y is 6 + (10a + 4), which simplifies to 10a + 10, and is clearly divisible by 4.
Therefore, by considering both statements together, we can determine that the sum x + y is divisible by 4. Thus, the answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
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If x and y are positive integers, is the sum x + y divisible by 4 ?(1) When the sum23x + 25y is divided by 10, the remainder is 8.(2) When22y 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 askedb)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question askedc)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficientd)EACH statement ALONE is sufficient to answer the question askede)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are neededCorrect answer is option 'C'. Can you explain this answer?
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If x and y are positive integers, is the sum x + y divisible by 4 ?(1) When the sum23x + 25y is divided by 10, the remainder is 8.(2) When22y 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 askedb)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question askedc)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficientd)EACH statement ALONE is sufficient to answer the question askede)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are neededCorrect answer is option 'C'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If x and y are positive integers, is the sum x + y divisible by 4 ?(1) When the sum23x + 25y is divided by 10, the remainder is 8.(2) When22y 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 askedb)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question askedc)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficientd)EACH statement ALONE is sufficient to answer the question askede)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are neededCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If x and y are positive integers, is the sum x + y divisible by 4 ?(1) When the sum23x + 25y is divided by 10, the remainder is 8.(2) When22y 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 askedb)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question askedc)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficientd)EACH statement ALONE is sufficient to answer the question askede)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are neededCorrect answer is option 'C'. Can you explain this answer?.
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