If x is the number
x=7*p+1
x=9*q+1
x=64*r+3
From the first 2 equations is obvious that
x=63*s+1
The number is
556*63+1 .....
634*63+1
simultaneously has to be
547*64+3 ...
624*64+3
My son wrote a 3 lines Python code and the result is 36163. But a math solution is needed
The key to the problem is to express $63$ as $(64 - 1).$
Then, $n$ must have form
$$63s + 1 = (64 - 1)s + 1 = 64s + (1 - s).$$
By the constraint of the problem, you then must have that
$$(1 - s) \equiv 3 \pmod{64} \implies s \equiv 62\pmod{64}.$$
So, it is simply a matter of finding the appropriate element $s$ from the set
$$\{62, (62 + [1 \times 64]), (62 + [2 \times 64]), \cdots \},$$
so that $n = 63s + 1$ is in the proper range, as specified by the constraint of the problem.
$$(63 \times 62) + 1 = 3907.$$
$$35000 - 3907 = 31093.$$
$$40000 - 3907 = 36093.$$
Each time that $s$ is replaced by $s + 64$, the product $63(s)$ changes to $63(s + 64)$, which is an increase of $(63 \times 64) = 4032.$
So, you must increase $s$ from $(62)$ to $62 + a(64),$ where the variable $a$ is chosen so that
$$31093 \leq 63(a \times 64) \leq 36093.$$
Notice that:
$$31093 = (4032 \times 7) + 2869.$$
$$36093 = (4032 \times 8) + 3837.$$
So, you want $$s = 62 + (8 \times 64) = 574.$$
This implies that
$$n = (63s + 1) = (63 \times 574) + 1 = 36163.$$