Is The Sum Of Four Different Positive Prime Integers An Odd Integer? GMAT Data Sufficiency

Question: Is the sum of four different positive prime integers an odd integer?

(1) One of the integers is 11.
(2) One of the integers is 2.

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“Is the sum of four different positive prime integers an odd integer?” - is a topic of the GMAT Quantitative reasoning section of GMAT. GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solutions and Explanation

Approach Solution : 1

The main idea here is that there is only one EVEN prime number which is 2, and every other prime number is ODD.
Therefore, if two is one of the four prime numbers, the sum equals EVEN + ODD + ODD = ODD.
The sum is ODD + ODD + ODD + ODD = EVEN if 2 is not one of the four prime numbers.
Therefore, 2 must be one of the four prime numbers for the sum to be ODD.
Let the rephrased target question be, Is number two one of the four primes?

Statement - 1: 11 is one of the integers

Since we don't know anything about the other three primes, number two might or might not be one of them.
We are unable to definitively respond to the rephrased target question
As a result, this statement is not sufficient

Statement - 2: 2 is one of the integers

It is given that 2 is one of the four prime numbers
Now, we have certain answer to the rephrased target question
As a result, This statement is sufficient

Correct Answer: (B)

Approach Solution : 2

The following points are the concept here:

• There is only one EVEN prime number, which is 2. The remaining prime numbers are ODD.
• If the sum of four odd integers is even, then at least one of the four prime numbers must be two.

Statement - 1: 11 is one of the integers

The prime numbers may be {2, 3, 5, 11}
Their total sum is ODD
The prime numbers may be {3, 5, 7, 11}
Their total sum is EVEN
As a result, this statement is not sufficient

Statement - 2: 2 is one of the integers

Now that the final three primes must be odd, the sum of the four primes must also be odd
As a result, this statement is sufficient

Correct Answer: (B)

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