【Question 14】In the book "Xu Gu Zhai Qi Suan Fa" written by the Southern Song mathematician Yang Hui, there is such a problem: “When divided by 2, the remainder is 1; when divided by 5, the remainder is 2; when divided by 7, the remainder is 3; when divided by 9, the remainder is 4. What is the original number?
In modern language, it can be stated as: “There is a number that, when divided by 2, leaves a remainder of 1; when divided by 5, leaves a remainder of 2; when divided by 7, leaves a remainder of 3; and when divided by 9, leaves a remainder of 4. What is this number?” Please write the smallest natural number that satisfies these conditions here $\qquad$
Analysis:
(Solution 1)
First, consider the number that leaves a remainder of 2 when divided by 5, a remainder of 3 when divided by 7, and a remainder of 4 when divided by 9.
Using the Chinese Remainder Theorem, we get
$$
\begin{array}{l}
5 \times 7 \times \quad+5 \times 9 \times \quad+7 \times 9 \times \\
5 \times 7 \times 5+5 \times 9 \times 1+7 \times 9 \times 4=472 \\
{[5,7,9]=315}
\end{array}
$$
$+5 \times 9 \times$ $+7 \times 9 \times$ $\qquad$
Therefore, $472 \pm 315 k$ all satisfy the conditions of leaving a remainder of 2 when divided by 5, a remainder of 3 when divided by 7, and a remainder of 4 when divided by 9.
The smallest is 472-315=157, and it also satisfies the condition of leaving a remainder of 1 when divided by 2.
(Solution 2)
Numbers that leave a remainder of 1 when divided by 2 are: $1,3,5,7,9,11,13,15,17, \cdots$;
Numbers that leave a remainder of 2 when divided by 5 are: $2,7,12,17 \cdots$;
Numbers that leave a remainder of 3 when divided by 7 are: $3,10,17 \cdots$;
So the numbers that satisfy "leaving a remainder of 1 when divided by 2, a remainder of 2 when divided by 5, and a remainder of 3 when divided by 7" are of the form $[2,5,7] n+17=70 n+17$ (where $n$ is a natural number). At this point, we only need to find the smallest $n$ that satisfies the condition of leaving a remainder of 4 when divided by 9.
When $n=2$, it satisfies the condition of leaving a remainder of 4 when divided by 9, so the smallest natural number that satisfies the conditions is $70 \times 2+17=157$.