There are four consecutive two-digit positive integers; their sum is a two-digit integer and a multiple of 11.

Let the numbers be x, x+1, x+2 and x+3

The sum of the 4 numbers is x + x+1 + x+2 + x+3 = 4x + 6

The largest two digit multiple of 11 is 99. But if `4x + 6 = 99 => 4x = 93` which gives a non-integer value for x.

The multiple of 11 smaller than 99 is 88.

4x + 6 = 88

=> 4x = 82

82 also is not a multiple of 4

The multiple 77 does satisfy the conditions.

4x + 6 = 66

=> 4x = 60

=> x = 15

The four numbers are 15, 16, 17 and 18. The product of the four integers is 15*16*17*18 = 73440

**The largest possible product of the 4 numbers is 73440**