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From the integers 1, 2, 3,..., 2004, two distinct numbers x and y are selected at random without replacement. The probability that x³ + y³ is divisible by 3 equals
- 1/3
- 2/3
- 1/6
- 1/4
Correct answer: 1/3
Solution
The sum of cubes, x³ + y³, is divisible by 3 if both x and y have the same remainder when divided by 3. Since the integers 1 to 2004 yield three possible remainders (0, 1, 2), the probability of selecting two numbers with the same remainder is calculated by considering the combinations of pairs from each remainder class, leading to a final probability of 1/3.
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