Notice that all prime numbers are odd, except the number 2. (All larger even numbers are divisible by 2, so they cannot be prime.) Thus, the sum of any two primes will be even (“Add two odds …”), unless one of those primes is the number 2. So, if you see a sum of two primes that is odd, one of those primes must be the number 2. Conversely, if you know that 2 cannot be one of the primes in the sum, then the sum of the two primes must be even.
If a and b are both prime numbers greater than 10, which of the following may be true? Indicate all that apply.
- ab is an even number.
- The difference between a and b equals 117.
- The sum of a and b is even.
Because a and b are both prime numbers greater than 10, they must both be odd. Therefore, ab must be an odd number, so choice (A) cannot be true. Similarly, if a and b are both odd, then a − b cannot equal 117 (an odd number). This difference must be even. Therefore, choice (B) cannot be true. Finally, because a and b are both odd, a + b must be even, so choice (C) will always be true.
The difference between the factors of prime number x is 1. The difference between the factors of prime number y is 2. Is xy even?