If the sum of a set of ten different positive prime numbers is an even

Question

If the sum of a set of ten different positive prime numbers is an even number, which of the following prime numbers CANNOT be in the set?

A. 2
B. 3
C. 5
D. 7
E. 11

PS21157

Zarrolou · Answer

Every prime number except 2 is odd.

If we sum  10  odd numbers we get an  even  number.
If we sum  9  odd numbers and  1  even number we get an  odd  number.

Since the sum is  even , all primes must be  obb  => no 2.

grotten · Answer

The set contains 10 numbers. Divide the 10 numbers in sets of two numbers.
Knowing that Odd+Odd = Even
Then the 5 - 2 number set would be:

(Odd + Odd) + (Odd + Odd) + (Odd + Odd) + (Odd + Odd) + (Odd + Odd) = Even + Even + Even + Even + Even = Even !! 
This concludes that every number in the set must be Odd, because Odd + Even = Odd
The only even number is 2.
 OA (A)

TeamGMATIFY · Answer

All prime numbers apart from 2 are odd.
Even + Even = Even
Odd + Even = Odd
Odd + Odd = Even

We are given ten different prime numbers, whose sum is even
If we include 2, we will have 9 odd prime numbers and one even.
This sum would be odd

If we exclude 2, we will have 10 odd numbers.
This sum would be even

Hence 2 is not included.
Option A

BrentGMATPrepNow · Answer

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

The key concept here is that 2 is the only EVEN prime number. All other primes are ODD. 
Since the set contains 10  different  prime numbers , there are only two possible cases: 
case 1) 2 is in the set of primes
case 2) 2 is NOT in the set of primes

case 1: If 2 IS in the set, then the set contains 9 ODD primes and 1 EVEN prime
In this case, the sum of the 10 primes will be ODD. 
However, the question tells us that the sum is EVEN. 
So, it CANNOT be the case that 2 is in the set of numbers.

Answer: A

Cheers,
Brent

ScottTargetTestPrep · Answer

Consider odd + odd = even, but odd + odd + odd = odd.

Now, if there are 10 odd numbers, the sum will be even because each pair of odds makes an even sum. But if there are 9 odd numbers and 1 even number, then the sum of the 9 odds will be odd, and adding that one even number will make the sum of those 10 numbers odd, because odd + even = odd.

Since we are told that the sum of 10 numbers is even, and since all primes are odd except 2, then 2 cannot be in the set; otherwise, the sum would be odd and not even.

Answer: A

DanTheGMATMan · Answer

2 is the only even prime, everything else is odd plus odd. Also important that there are an even number of terms in the sum and they are all distinct so 2 can't be involved, otherwise the sum would be odd:

https://www.youtube.com/watch?v=7geICjSguKY

bumpbot · Answer

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If the sum of a set of ten different positive prime numbers is an even

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If the sum of a set of ten different positive prime numbers is an even

Prep Toolkit

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