5. (6 points) A number when divided by 5 leaves a remainder of 2, when divided by 7 leaves a remainder of 3, and when divided by 11 leaves a remainder of 7. The smallest natural number that satisfies these conditions is $\qquad$ 227
【Solution】Solution: Numbers that are "divisible by 5 with a remainder of 2" are $7,12,17,22,27,32,37,42,47,52 \cdots$, numbers that satisfy "divisible by 7 with a remainder of 3" are $10, 17, 24, 31, 38, 45, 52$, we can find 52; numbers that simultaneously satisfy "divisible by 5 with a remainder of 1" and "divisible by 7 with a remainder of 3" differ by multiples of $5 \times 7=35$, such as 52,

Next, find numbers that satisfy "divisible by 11 with a remainder of 7", we can find 227. Since $227<[5,7,11]=505$, the smallest natural number sought is 227.

Therefore, the answer is: 227.