Question 190369: # 5. Find a counterexample to show that the following statement is incorrect: “The sum of any two prime numbers is divisible by 2”
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! # 5
Q: Find a counterexample to show that the following statement is incorrect: “The sum of any two prime numbers is divisible by 2”
A:
Recall that a prime number is a number that is ONLY divisible by 1 and itself. So the first few prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc...
Notice how every prime number (except 2) is an odd number. Now if we select two random prime numbers, there's a good chance that they'll both be odd. For example, let's say I choose 17 and 31. These are both odd numbers. If I add them, I get
17+31=48
Since  , this means that 48 is divisible by 2. It turns out that the sum of ANY two odd numbers is ALWAYS even. So there's a good chance that the sum of any two prime numbers is even (since all of them, but the number 2, are odd)
So to find a counterexample, we MUST select 2 as one of the numbers, since it is only even prime. The other number can be odd since the sum of an even and odd number is ALWAYS odd.
So I'm going to select the numbers 2 and 3 (both are prime, but the first one is even and the second is odd)
So 2+3=5 which is NOT divisible by 2 since 
So this shows that the statement “The sum of any two prime numbers is divisible by 2” is false. We've shown this by adding the prime numbers 2 and 3 to get 5 (which is NOT divisible by 2).
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