[1] What are the smallest 5 possible values of b?
[2] What is the largest value b cannot assume?

Now let us generalize the second part.

[3] Find the largest value of b where a/b cannot equal b when it is multiplied by 10^n and rounded to the nearest whole number. (Part [2] is n=2)
Solve for n = 1, 2, 3, 4, ...

Part 1:

The five smallest possible values of b are 10, 14, 20, 30 and 33, as shown in the table below for the 47 smallest possible values of b.

DEFDBL A-Z

FOR b = 1 TO 10000
  a0 = b * b / 100
  a1 = INT(a0) ' int is floor function
  a2 = -INT(-a0)  ' this results in ceiling
  p1 = INT(100 * a1 / b + .5)
  p2 = INT(100 * a2 / b + .5)
  IF p1 = b THEN a = a1:  ELSE IF p2 = b THEN a = a2:  ELSE a = -1
  IF a > 0 THEN
    IF ct < 47 THEN PRINT a; b, 100 * a / b
    ct = ct + 1
  ELSE
    badB = b
  END IF
NEXT b
PRINT badB

 a   b             %
 1  10        10
 2  14        14.28571428571429
 4  20        20
 9  30        30
11  33        33.33333333333334
13  36        36.11111111111111
16  40        40
17  41        41.46341463414634
21  46        45.65217391304348
22  47        46.80851063829788
23  48        47.91666666666666
24  49        48.97959183673469
25  50        50
26  51        50.98039215686274
27  52        51.92307692307692
28  53        52.83018867924528
29  54        53.7037037037037
30  55        54.54545454545455
35  59        59.32203389830509
36  60        60
37  61        60.65573770491803
40  63        63.49206349206349
41  64        64.0625
42  65        64.61538461538461
45  67        67.16417910447761
46  68        67.64705882352941
49  70        70
52  72        72.22222222222223
53  73        72.60273972602739
55  74        74.32432432432432
56  75        74.66666666666667
58  76        76.31578947368421
59  77        76.62337662337663
61  78        78.2051282051282
64  80        80
66  81        81.48148148148148
67  82        81.70731707317073
69  83        83.13253012048193
72  85        84.70588235294117
74  86        86.04651162790698
76  87        87.35632183908046
77  88        87.5
79  89        88.76404494382022
81  90        90
83  91        91.20879120879121
85  92        92.39130434782609
88  94        93.61702127659575

Part 2:

The above program also produces the answer to part 2:

93 

Beginnings of part 3:

b
1             6
2             93
3             950
4             9842
5             99500
6             999293
7             9997764
8             9999999

The below program was stopped at this point as the answers had become meaningless as they had become limited by the internal limitations of the program, instituted for time purposes.

DEFDBL A-Z
CLS

mult = 10
FOR pwr = 1 TO 20
FOR b = 1 TO 10000000
  a0 = b * b / mult
  a1 = INT(a0) ' int is floor function
  a2 = -INT(-a0)  ' this results in ceiling
  p1 = INT(mult * a1 / b + .5)
  p2 = INT(mult * a2 / b + .5)
  IF p1 = b THEN a = a1:  ELSE IF p2 = b THEN a = a2:  ELSE a = -1
  IF a > 0 THEN
    REM nothing here
  ELSE
    badB = b
  END IF
NEXT b
PRINT pwr, badB
mult = mult * 10
NEXT pwr