False.
To justify the answer, it is necessary to solve the congruence relation 5636^17 ≡1 mod 9.
First, it's recognized that for any positive integer n, n^1 is congruent to n mod 9. This step is fairly straightforward, as it simply states that for all positive integers n, the number n will give the same remainder as n when divided by 9. It also has to be observed that any number can be expressed in mod 9 using its digit sum due to mathematical property of 9.
Taking 5636 mod 9, we find that the remainder r when 5636 is divided by 9 is equivalent to summing its digits and taking mod 9. This value of r is the one that will be used for further calculations.
Then, it's necessary to check whether r to the power of 17 is congruent to 1 mod 9. This is based on the fact that if 5636^17 is congruent to 1 mod 9, then r^17 should also be congruent to 1 mod 9.
So, the congruence r^17 is computed and then checked if it's equivalent to 1 modulo 9.
If the congruence equation holds true, then there exists an integer r such that the equation 5636^17 ≡1 mod 9 holds true.
However, after performing these calculations, we find that r^17 is not congruent to 1 mod 9. Therefore, the statement that there exists an integer r such that 5636^17 ≡1 mod 9 is false.