Vincen 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 on the set?
A. 2
B. 3
C. 5
D. 7
E. 11
The OA is the option A.
Why 2 can't be on the set? Should I try making a set of 10 prime numbers? Is there an easier way to solve this PS question?
Prime numbers are odd numbers.
The sum of a pair of odd numbers is an even number. The sum of even numbers is an even number.
The sum of three numbers that are even, even and odd is an odd number. For example, 2, 98 and 29 = 127 is odd.
Assume the first prime is 2. That leaves 9 other primes to be added together, which is the same thing as 4 pairs of primes added to 1 additional prime.
So, adding the ten primes including 2 would be to add 2, 4 pairs of primes, and another prime.
2 is even. 4 pairs of primes is even because a pair of primes adds to an even number and even numbers added together is even. Finally, the last prime is an odd number.
Referring to the sum of three numbers rule above, the sum of these numbers is therefore odd, which contradicts the problem statement.
Therefore, 2 can't be the first prime.
