x = 3 mod 17,
x = 4 mod 11 (since 6 pirates were killed),
x = 5 mod 6.

By the first congruence, it follows that x = 3 + 17a for some integer a. So 3 + 17a = 4 mod 11. Hence, 17a = 1 mod 11.

Thus, a = ¹/₁₇ mod 11 = 2 mod 11 [since (17)(2) = 34 = 1 mod 11].

So a = 2 + 11b for some integer b. Whence, x = 3 + 17a = 3 + 17(2 + 11b) = 3 + 34 + 187b = 5 mod 6.

So x = 37 + 187b = 5 mod 6;

32 + 187b = 0 mod 6;

32 + 187b = 36 mod 6;

187b = 4 mod 6.

Note that 187 = 1 mod 6. So we have b = 4 mod 6. I.e., b = 4 + 6c for some integer c. Since b is positive, it follows that b > 4. Therefore, x > 37 + 187(4) = 37 + 748 = 785.

So the cook recieves at least 785 coins.