Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3, \dots, 2010\}$. What is the probability that $abc ab a$ is divisible by 3

The probability that abc ab a is divisible by 3 is 13/27.

We group this into groups of 3, because 3|2010. This means that every residue class mod 3 has an equal probability.

If 3|a, we are done. There is a probability of 1/3 that that happens.

Otherwise, we have 3 | bc + b + 1, which means that b (c + 1 ) = 2 mod{3}. So either

b = 1 mod (3) c = 1 mod (3)

or

b = 2 mod (3) c = 0 mod (3)

Which will lead to the property being true. There is a 1/3 * 1/3 = 1/9 chance for each bundle of cases to be true.

Thus, the total for the cases is 2/9. But we have to multiply by 2/3 because this only happens with a 2/3 chance. So the total is actually 4/27.

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