If the number 583a4a is divisible by 6, then what is the sum of all the possible values of a?

For a number to be divisible by 6, it must be simultaneously divisible by both 2 and 3.
Divisibility by 2:
The last digit of the number 583a4a is a. For the number to be even (divisible by 2), a must be an even digit:

a∈{0,2,4,6,8}
Divisibility by 3:
The sum of the digits of the number must be divisible by 3.

Sum of digits=5+8+3+a+4+a=20+2a
Let's test the possible even values of a:

If a=0: Sum=20+2(0)=20 (not divisible by 3)

If a=2: Sum=20+2(2)=24 (divisible by 3 ✓)

If a=4: Sum=20+2(4)=28 (not divisible by 3)

If a=6: Sum=20+2(6)=32 (not divisible by 3)

If a=8: Sum=20+2(8)=36 (divisible by 3 ✓)

The possible values of a are 2 and 8.

Sum of all possible values=2+8=10