If a, b, and c are integers and a + bc is even, what is the remainder

Question

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.

gmatophobia · Answer

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  
eq  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

SaquibHGMATWhiz · Answer

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: nature.png 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

bumpbot · Answer

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If a, b, and c are integers and a + bc is even, what is the remainder

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