There are four consequtive two-digit number positive integers; their sum is a two-digit integer and a multiple of 11. What is the largest possible product of these four numbers? Therefore, what is the smallest positive integer which is divisible by 11 and sum of all digits equal 13?

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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

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