To solve the problem, we start by noting that (a) and (b) are multiples of 3. This means we can express them as:
(a = 3k) and (b = 3m) (where (k) and (m) are integers).

Step 1: Compute (a^2 + b^2)

Substitute (a = 3k) and (b = 3m) into the expression:
[ a^2 + b^2 = (3k)^2 + (3m)^2 = 9k^2 + 9m^2 = 9(k^2 + m^2) ]

Step 2: Divide by 3

The expression (9(k^2 + m^2)) can be rewritten as (3 \times [3(k^2 + m^2)]), which is clearly a multiple of 3.

When a multiple of 3 is divided by 3, the remainder is 0.

Answer: (\boxed{0})

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