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ANSWER: He took 2 bags with 16 coins and 4 bags with 17 coins.

EXPLANATION: The table below shows the maximum number of each bag that could have been taken:


If you add up all the possible totals from the different combinations of 16-coin bags and 17 coin bags, you get the following 25 possible totals:

0
16
17
32
33
34
48
49
50
51
61
64
66
67
68
81
82
83
84
85
96
97
98
99
100 (2 bags of 16 coins and 4 bags of 17 coins)

If you add up all the possible totals from the different combinations of 23-coin bags, 24-coin bags, 39-coin bags, and 40-coin bags, you get the following 25 possible totals:

0
23
24
46
47
48
62
63
64
69
70
71
72
78
79
80
85
86
87
88
92
93
94
95
96

Finally, if you add up all the different pairs of totals from each of those 2 lists, only one gives a total of 100 coins (2 bags of 16 coins and 4 bags of 17 coins).

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