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# Is the sum of four particular integers even? (1) Two of the integers

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Is the sum of four particular integers even? (1) Two of the integers  [#permalink]

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26 Apr 2019, 02:01
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Is the sum of four particular integers even?

(1) Two of the integers are odd and two are even.
(2) The average (arithmetic mean) of the four integers is an integer.

DS21602.01
OG2020 NEW QUESTION

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Re: Is the sum of four particular integers even? (1) Two of the integers  [#permalink]

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26 Apr 2019, 02:16
3
To get the sum of four integers even, several cases are possible
1) all of them are even
2) There are even number of odd integers

1 stm: odd+odd+even+even=even. Suff
2 stm: $$\frac{sum of four integers}{4}$$=integer --> sum of four integers=4*integer. suff

IMO
Ans: D
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Re: Is the sum of four particular integers even? (1) Two of the integers  [#permalink]

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27 Apr 2019, 05:47
1
Bunuel wrote:
Is the sum of four particular integers even?

(1) Two of the integers are odd and two are even.
(2) The average (arithmetic mean) of the four integers is an integer.

DS21602.01
OG2020 NEW QUESTION

#1
o+o+e+e= e
sufficient
#2
avg of 4 integers is integers means sum of 4 intgers is even
sufficient
IMO D
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Re: Is the sum of four particular integers even? (1) Two of the integers  [#permalink]

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28 Apr 2019, 07:36
Top Contributor
Bunuel wrote:
Is the sum of four particular integers even?

(1) Two of the integers are odd and two are even.
(2) The average (arithmetic mean) of the four integers is an integer.

Some important rules:
#1. ODD +/- ODD = EVEN
#2. ODD +/- EVEN = ODD
#3. EVEN +/- EVEN = EVEN

#4. (ODD)(ODD) = ODD
#5. (ODD)(EVEN) = EVEN
#6. (EVEN)(EVEN) = EVEN

Target question: Is the sum of four particular integers even?

Statement 1: Two of the integers are odd and two are even.
We have: SUM = ODD + ODD + EVEN + EVEN
= (ODD + ODD) + (EVEN + EVEN)
= EVEN + EVEN
= EVEN
The answer to the target question is YES, the sum is even
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The average (arithmetic mean) of the four integers is an integer.
Let the 4 numbers be a, b, c, and d
We can write: (a + b + c + d)/4 = k, where k is some integer
So, we can also write: a + b + c + d = 4k
Since 4k is EVEN, we know that a + b + c + d is EVEN
The answer to the target question is YES, the sum is even
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
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Re: Is the sum of four particular integers even? (1) Two of the integers  [#permalink]

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28 Apr 2019, 12:56
Hi All,

We're told that we have four INTEGERS. We're asked if the sum of those four integers is EVEN. This is a YES/NO question and can be solved with either Number Property rules or TESTing VALUES.

(1) Two of the integers are ODD and two are EVEN.

Fact 1 gives us enough information to deal with some specific Number Properties.
Adding two ODD numbers will give us a sum that is EVEN (for example, 3+5 = 8)
Adding two EVEN numbers will give us a sum that is EVEN (for example, 2+4 = 6)

Thus, adding two ODDs and two EVENs will give us EVEN+EVEN = EVEN. The answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT

(2) The average (arithmetic mean) of the four integers is an integer.

Fact 2 references the 'Average Formula', so we should think in those terms. If we refer to the values as A, B, C and D, we can set up an average as:

(A+B+C+D)/4 = Integer
(A+B+C+D) = (4)(Integer)

From this equation, we know that the sum of the four integers will be 4 times an integer. Multiplying ANY integer by an EVEN number will give us a product that is EVEN. For example:
4(-1) = -4
4(0) = 0
4(1) = 4
4(2) = 8
4(3) = 12
Etc.
Thus, the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT

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Re: Is the sum of four particular integers even? (1) Two of the integers  [#permalink]

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06 May 2019, 20:00
Bunuel wrote:
Is the sum of four particular integers even?

(1) Two of the integers are odd and two are even.
(2) The average (arithmetic mean) of the four integers is an integer.

DS21602.01
OG2020 NEW QUESTION

Statement One Alone:

Two of the integers are odd, and two are even.

Since we have two even and two odd integers, we have:

odd + odd + even + even = even + even = even

Statement one alone is sufficient to answer the question.

Statement Two Alone:

The average (arithmetic mean) of the four integers is an integer.

Since the average of the 4 numbers is an integer:

sum/4 = integer

sum = 4 * integer

Thus, we see that the sum must be even.

Statement two alone is sufficient to answer the question.

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Is the sum of four particular integers even? (1) Two of the integers  [#permalink]

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14 May 2019, 05:42
1

Solution

Steps 1 & 2: Understand Question and Draw Inferences

In this question, we are asked to determine
• Whether the sum of four particular integers even or not.

To determine this, we need to know either the values of the integers, or the even-odd nature of them.
With this understanding, let us now analyse the individual statement.

Step 3: Analyse Statement 1

As per the information given in statement 1, two of the integers are odd and two are even.
• Therefore, the sum of the four integers = (odd + odd) + (even + even) = even + even = even
As we can determine that the sum of the four integers is even, statement 1 is sufficient to answer the question.

Step 4: Analyse Statement 2

As per the information given in statement 2, the average (arithmetic mean) of the four integers is an integer.
• Hence, we can say, sum of the four integers = average of the four integers × 4
• As any multiple of 4 is an even number, the sum of the four integers is also an even number
Hence, statement 2 is sufficient to answer the question.

Step 5: Combine Both Statements Together (If Needed)

Since we can determine the answer from either of the statements individually, this step is not required.
Hence, the correct answer choice is option D.

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Is the sum of four particular integers even? (1) Two of the integers   [#permalink] 14 May 2019, 05:42
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