Mirror Multiples

Extensions:

1. Find the smallest such integer.
2. Can such an integer be found for multiples 2 thru 9?
3. Does such an integer exist for any number of digits in the integer? Find examples for different numbers of digits.

Source: Original.

Above all, Philippe gets high marks for the briefest solution: "I've already solved this for your previous Reversed Products puzzle." (Radu is also listed there as a solver.) My apologies for reusing a puzzle - I didn't even realize I was doing it! - KD.

A good new solution was received from Larry Baum:

1) 1089 (9801/1089 = 9)

2) No, only for 4 or 9

Let i = x....y be the integer of interest (x is leading digit and y is
unit digit) and d be the integer multiple we are trying to achieve; i.e.
d*i = reverse i.

We must have:
a) d*y = x mod 10
b) d*x < 10
c) x > 0 (i.e the reverse of 123 is NOT 3210 or 32100)

b) => if d > 4, x must be 1 => (by a) d is odd

If d = 5, c => y is odd => x = 5 but that violates b)
If d = 7, a => y = 3, i = 1abc3.  But 7*1abc3 > 70000 so cannot be 3cba1

If d = 2, x is even < 5, i.e. x = 2 or 4.  But 2*2abcy = 4zzzz or 5zzzz
=> y = 4 or 5; but neither 2*4 nor 2*5 = 2 mod 10

If d = 3, x=1,2 or 3.  
If x=1, a=> y=7, but 3*1zzz7 < 60000 & cannot start with 7
If x=2, a=> y=4, but 3*2zzz4 > 60000 & cannot start with 4
If x=3, a=> y=1, but 3*3zzz1 > 90000 & cannot start with 1

Thus the only possible d's are 4 and 9.
9*1089 = 9801
4*2178 = 8712

3) yes, not number of digits = 4 or more

You can insert as many 9's as you like in the middle of 1089 or 2178 and
stil have a reverse multiple; e.g.:

9*10999999999989 = 98999999999901
4*21999978 = 87999912

If you allow for a leading and a trailing 0, then you
can find a solution for a ratio of 6:

5604390 / 934065 = 6