Suppose such an integer with sum of digits equals 76 exists. Note that 111111111111 is divisible by 37, so we can reduce each digit by 1, the number now have the following property: It consists only of digits 4 and 8, at most 12 digit, and the sum of digits equals 64. Since $\gcd(4,37)=1$, we can divide the integer by 4, so it consists only of digits 1 and 2, the sum of digits is 16. Note that $10^3\equiv1\pmod{37}$. So if we divide the digits into four groups of 3 consecutive digits, the sum of these 3-digit numbers is still divisible by 37. This integer is 7 mod 9 and 0 mod 37, so it must be 259, 592, or 925. But the largest possible digit is 8, we get a contradiction.