3. (10 points) A certain three-digit number is a multiple of 2, adding 1 makes it a multiple of 3, adding 2 makes it a multiple of 4, adding 3 makes it a multiple of 5, and adding 4 makes it a multiple of 6. What is the smallest number that satisfies these conditions? $\qquad$ .
【Analysis】A multiple of 2 is an even number. We find that adding 1 is a multiple of 3, adding 2 is a multiple of 4, adding 3 is a multiple of 5, and adding 4 is a multiple of 6. This means the number leaves a remainder of 2 when divided by 3, 4, 5, and 6. This is a congruence problem.

【Solution】Solution: An even number leaves a remainder of 2 when divided by 3, 4, 5, and 6.
Therefore, this three-digit number minus 2 is a common multiple of 3, 4, 5, and 6.
The least common multiple of 3, 4, 5, and 6: \(3 \times 4 \times 5 = 120\), \(120 + 2 = 122\).
Thus, the answer is 122.