3. (8 points) A positive integer greater than 1, when increased by 1, is divisible by 2; when increased by 2, is divisible by 3; when increased by 3, is divisible by 4; when increased by 4, is divisible by 5. The smallest positive integer is $\qquad$ .
3. (8 points) A positive integer greater than 1, when increased by 1, is divisible by 2; when increased by 2, is divisible by 3; when increased by 3, is divisible by 4; when increased by 4, is divisible by 5. The smallest positive integer is $\qquad$ .

【Solution】Solution: According to the analysis:
This number, when divided by $2,3,4,5$, leaves a remainder of 1. Therefore, this number minus 1 can be simultaneously divisible by $2,3,4,5$; the least common multiple of $2,3,4,5$ is 60, so this number is a multiple of 60 plus 1.

Since this number is greater than 1, the smallest number is 61.
Therefore, the answer is: 61.