If a, b, and c are three different positive integers whose sum is prime, which of the following statements could be true? Answers:
A) Each of a, b, and c is prime.
B) Each of a + 3, b + 3, and c + 3 is prime
C) Each of a + b, a + c, and b + c is prime
D) The average (arithmetic mean) of a, b, and c is prime
E) a + b = c
3 different +ve integers (assume a, b, c) – sum is a prime number   All prime numbers except 2 are odd.   Consider 2 -&gt; There is no combination of a, b &amp; c to get a sum of 2. (a, b and c have to be different and positive integers)   Now for a, b and c to add up to a prime number (&gt;2) the possibilities are   2 numbers are even positive integers and 1 is an odd positive integer ( E, E, O – i.e. assume a &amp; b are even and c is odd)   OR   All 3 are odd positive integers (O, O, O – a, b &amp; c are odd)   (A)	E, E, O (even if 1 of the even numbers is 2 there is another number which cannot be prime) combination not true   but O, O, O combination could be true.   (B)	E+3 could be prime but O+3 is an even integer &gt;2 so cannot be prime   So E,E,O combination not true (a+3 OK, b+3 OK but c+3 – cannot be prime ) and O,O,O (a/b/c+3 cannot be prime)combination is also not true.   (C)	E+O could be prime but O+O cannot be So E,E,O (a+b cannot be prime) combination and O,O,O (a+b, b+c and c+a cannot be prime) combination not true   (D)	If the average of 3 numbers is a prime number then the sum of the 3 numbers is a multiple of 3 and the arithmetic mean. So the sum cannot be even – therefore this option is also not possible.   (E)	If a+b=c then a+b+c = 2c which is an even no and hence this option is not possible   Each of a, b, and c is prime.nswer (Each of a, b, and c is prime.)
<result>A</result>
If a, b, and c are three different positive integers whose sum is prime, which of the following statements could be true?