Answer: The integer is 37.
Given:The given integer leaves remainders 1, 2, and 6 when divided by 9, 11, and 13, respectively.
Explanation: To find this integer, we can use the Chinese Remainder Theorem. Let the integer be x, then:
x ≡ 1 (mod 9)
x ≡ 2 (mod 11)
x ≡ 6 (mod 13)
Applying the Chinese Remainder Theorem, we find that the smallest positive solution is x = 37.
Since the integer is between 1 and 1200, the possible values of x are 37, 1294, 2551, and so on.
The only value within the given range is 37.
So, the integer is 37.