If a,a+2,a+4 are all prime numbers, then the number of possible values of a is

Let's examine the set of numbers {a,a+2,a+4} where a must be a natural number:
Among any three consecutive odd numbers, exactly one must be a multiple of 3.
For all three numbers to be prime simultaneously, the one that is a multiple of 3 must equal 3 itself (since 3 is the only prime multiple of 3).

Let's test this condition:
If a=3:

a=3 (Prime)

a+2=3+2=5 (Prime)

a+4=3+4=7 (Prime)

This gives us the valid prime triplet {3,5,7}.
If we pick any other value for a, one of the numbers in the triplet will be a composite multiple of 3 (for example, if a=5, the triplet is {5,7,9}, where 9 is composite).

Thus, a=3 is the only valid solution, meaning there is exactly one possible value for a.