The probability that a randomly chosen integer A, between 10 and 1000, will be divisible by a second integer B, formed by all but the last digit of A, is 123/991 (almost 1 in 8).
There are 991 possible values for A
If the last digit of A is not zero, the result of dividing A by B will be a quotient greater than 10. For example, if A is 22, B is 2, and the quotient is 11. All two-digit multiples of 11 (11, 22, 33, 44, etc.) have this property; there are 9 such multiples.
If A contains more than two digits, it is easy to see that
In addition to the first 9 multiples of 11, the first 5 multiples
of 12 (12, 24, 36, 48, and 60) are evenly divisible by their first digits
(but 60 has already been counted, since it ends in 0). Similarly, the
first 3 multiples of 13 (13, 26, and 39), and the first 2 multiples of
14 (14 and 28) are "good" values. Finally, the 5 remaining two-digit
numbers beginning with 1 (15, 16, 17, 18, and 19), but not their
multiples, are "good" numbers. In all, there are