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

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Re: If the sum of a set of ten different positive prime numbers is an even [#permalink]
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grotten wrote:
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

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
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Re: If the sum of a set of ten different positive prime numbers is an even [#permalink]
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grotten wrote:
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

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.

Cheers,
Brent
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Re: If the sum of a set of ten different positive prime numbers is an even [#permalink]
grotten wrote:
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

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.

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Re: If the sum of a set of ten different positive prime numbers is an even [#permalink]