The smallest positive integer

The numbers that satisfy condition IV form a proper subset of those that satisfy condition I, so we can just focus on the last three conditions.

Now to satisfy conditions II and III we are looking for integers $m,n$ such that

$5m + 2 = 7n + 3 \Longrightarrow 5m - 7n = 1,$

the least of which are $m = 3, n = 2.$ Thus $N = 17 + 35k$ for some integer $k.$

To satisfy conditions III and IV we are looking for integers $m,n$ such that

$7m + 3 = 9n + 4 \Longrightarrow 7m - 9n = 1,$

the least of which are $m = 4, n = 3.$ Thus $N = 31 + 63j$ for some integer $j.$

we are thus looking for $j,k$ that satisfy

$17 + 35k = 31 + 63j \Longrightarrow 35k - 63j = 14 \Longrightarrow 5k - 9j = 2,$

the least of which is $k = 4, j = 2,$ yielding $N = 17 + 35*4 = \boxed{157}.$

Let the natural number which will be multiplied by each number, $3, 5, 7, 9$ to $a$ .

Then $N = 3a+1 = 5a+2 = 7a+3 = 9a+4$ .

If $N$ is the multiplier of the upper expressions or can be divided by $3, 5, 7, 9$ , then it cannot satisfy the problem.

Then let's just find the number.

$9a+4$ always satisfies that the expression, $3a+1$ because $9 \times 5 + 1 = 46 = 3 \times 15 + 1$ , always satisfies.

Then we can just ignore it.

$N = 5a + 2 = 7a + 3 = 9a + 4$ .

The smallest positive integer $N$ is $\boxed{157}$ because $157 = 5 \times 31 + 2 = 7 \times 22 + 3 = 9 \times 17 + 4$ .

Hence, the answer is $\boxed{157}$ .