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If a, b, c, d are 4 non-negative integers. Is (a+c) even?

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If a, b, c, d are 4 non-negative integers. Is (a+c) even?  [#permalink]

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New post Updated on: 13 Aug 2018, 01:46
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If a, b, c, d are 4 non-negative integers. Is (a+c) even?

1. \(a^2 +b^2 + c^2\) is even
2. \(b^2 + c^2 + d^2\) is even

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient

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Originally posted by EgmatQuantExpert on 27 Feb 2018, 09:20.
Last edited by EgmatQuantExpert on 13 Aug 2018, 01:46, edited 2 times in total.
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Re: If a, b, c, d are 4 non-negative integers. Is (a+c) even?  [#permalink]

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New post 27 Feb 2018, 11:49
EgmatQuantExpert wrote:

Question:



If a, b, c, d are 4 non-negative integers. Is (a+c) even?

1. \(a^2 +b^2 + c^2\) is even
2. \(b^2 + c^2 + d^2\) is even

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient


we know

even + even = even
odd + odd = even

so

1) \(a^2 +b^2 + c^2\) is even now here a b c are even or 2 out a/b/c are odd and one is even
now a and b even c odd
a+c not even
now a and c odd b even
a+c even
insufficient

2) \(b^2 + c^2 + d^2\) is even
insufficient similar to
b c even d odd
we don't know value of a

now
combining
\(a^2 +b^2 + c^2\) + \(b^2 + c^2 + d^2\) = \(a^2 + 2 b^2 + 2 c^2 + d^2\)
here a^2 + d^2 is even as b/c can be odd or even and still the result is even

we know nothing about C being even/odd

(E) imo
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Re: If a, b, c, d are 4 non-negative integers. Is (a+c) even?  [#permalink]

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New post 27 Feb 2018, 12:11
We need to know the following odd-even properties
even^2 = even | odd^2 = odd | even + even = even | odd + odd = even | even + odd = odd

Is a+c even?

1. \(a^2 + b^2 + c^2\) is even
If \(a^2\), \(b^2\),and \(c^2\) are all even, then a+c is even
If \(a^2\) and \(b^2\) are odd and \(c^2\) is even, a+c is odd (Insufficient)

2. \(b^2 + c^2 + d^2\) is even
This statement contains no information about a.
We cannot say if a+c is even or not. (Insufficient)

On combining the information in both the statements,
We can have a case where a,b, and d are odd, whereas c is even. Here, a+c is odd
But, if a,b,c, and d are all even, then a+c will be even (Insufficient - Option E)
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Re: If a, b, c, d are 4 non-negative integers. Is (a+c) even?  [#permalink]

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New post 28 Feb 2018, 10:37

Solution:



Step 1: Analyse Statement 1:
\(a^2 +b^2 + c^2\) is Even
    • If \(a^2 +b^2 + c^2\) is Even, it gives rise to two possibilities:
      o Either one of \(a/b/c\) is Even and the rest two are Odd
      o Or, all the three numbers: \(a, b\) and \(c\)are Even.
Per our conceptual knowledge, let’s make a table to understand this.
Image
It is clear from the table that the even-odd nature of \((a + c)\)varies, and we cannot conclude anything about it.
As we do not know the exact even-odd nature of\((a + c)\),
Statement 1 alone is NOT sufficient to answer the question.
Hence, we can eliminate answer choices A and D.
Step 2: Analyse Statement 2:
\(b^2 +c^2 + d^2\) is Even
    • If \(b^2 +c^2 + d^2\) is Even, it gives rise to two possibilities:
      o Either one of \(b/c/d\) is Even and rest two are Odd
      o Or, all the three numbers: \(b, c\) and \(d\) are Even.
Let’s make a table to understand this.
Image
It is clear from the table that the even-odd nature of \((a + c)\) varies, and we cannot conclude anything about it.
As we do not know the exact even-odd nature of \((a + c)\),
Statement 2 alone is NOT sufficient to answer the question.
Hence, we can eliminate answer choice B.
Step 3: Combine both Statements:
From the first statement, we have: \(a^2 +b^2 + c^2\) = Even
From the second statement, we have: \(b^2 +c^2 + d^2\) = Even
Combining both the statements, we have: a\(^2 +b^2 + c^2 + b^2 +c^2 + d^2\) = Even + Even = Even
Therefore, \(a^2 +2b^2 + 2c^2 + d^2\) is even
We know that, \(2b^2 + 2c^2\) is always even irrespective of the even-odd nature of \(b\) and \(c\), since they are multiplied by an even number \((2)\)
So, our expression reduces to: \(a^2 + d^2\) is even. This will lead to two possibilities:
    • Either both a and d are Odd.
    • Or both a and d are Even.
Per our conceptual understanding, let us draw a table combining both the tables from Step 3 and Step 4 to see how will the even odd nature of the numbers affect the even-odd nature of the expression: \((a + c)\):
Image
From the table it is evident that the even-odd nature of the expression \((a + c)\) cannot be determined uniquely.
By combining both statements we did not get a unique answer.
Correct Answer: Option E
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Re: If a, b, c, d are 4 non-negative integers. Is (a+c) even? &nbs [#permalink] 28 Feb 2018, 10:37
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If a, b, c, d are 4 non-negative integers. Is (a+c) even?

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