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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
85%
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Question Stats:
56% (02:38) correct
44%
(02:35)
wrong
based on 112
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If a, b, and c are integers and a + bc is even, what is the remainder when a + c is divided by 2?
(1) a + b^2 is even.
(2) c^b + b^c leaves no remainder when divided by 2.
(1) a + b^2 is even.
(2) c^b + b^c leaves no remainder when divided by 2.
29 Sep 2023, 00:37
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RenB
If a, b, and c are integers and a + bc is even, what is the remainder when a + c is divided by 2?
(1) a + b^2 is even.
(2) c^b + b^c leaves no remainder when divided by 2.
(1) a + b^2 is even.
(2) c^b + b^c leaves no remainder when divided by 2.
Given
a + bc = even
Case 1: b ⇒ even
- a = even
- c = even or odd
Case 2: b ⇒ odd
- a = even
- c = even
Case 3: b ⇒ odd
- a = odd
- c = odd
Statement 1
(1) a + b^2 is even.
Let's assume
a = even, b = even
- If c = even → Remainder when a + c is divided by 2 = 0
- If c = odd → Remainder a + c is divided by 2 \(\neq\) 0
The statement alone is not sufficient. We can eliminate A and D.
Statement 2
(2) c^b + b^c leaves no remainder when divided by 2.
Assume a = b = even
From case 1 we know if b = even → a = even. Hence, we can conclude that (a+c) = even and is divisible by 2 and the remainder = 0
If c = b = odd
From case 3 we know if c & b = odd → a = odd. Hence, we can conclude that (a+c) = even and is divisible by 2 and the remainder = 0
This statement is sufficient.
Option B
29 Sep 2023, 01:57
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RenB
If a, b, and c are integers and a + bc is even, what is the remainder when a + c is divided by 2?
(1) a + b^2 is even.
(2) c^b + b^c leaves no remainder when divided by 2.
(1) a + b^2 is even.
(2) c^b + b^c leaves no remainder when divided by 2.
Solution:
Pre Analysis:
- According to the question, \(a + bc=even\) which is possible under four cases
- Those 4 cases and value of \(a+c\) in each are:
Attachment:
nature.png [ 14.54 KiB | Viewed 1567 times ]
- We are asked the even/odd nature of \(a+c\)
Statement 1: \(a + b^2\) is even
- \(a + b^2\) is even in cases 1 and 2 where \(a+c\) is even and odd respectively
- Thus, statement 1 alone is not sufficient and we can eliminate options A and D
Statement 2: \(c^b + b^c\) leaves no remainder when divided by 2
- According to this statement, \(c+b=even\) which is true in cases 1 and 4
- In both cases 1 and 4, nature of \(a+c\) is even
- Thus, statement 2 alone is sufficient
Hence the right answer is Option B
