I started by listing the two-digit numbers in each set to compare them.
Multiples of 5:
10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90
Triangular Numbers:
10, 15, 21, 28, 36, 45, 55, 66, 78, 91
Even numbers, excluding multiples of four:
10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98
Multiples of three, excluding multiples of nine:
12, 15, 21, 24, 30, 33, 39, 42, 48, 51, 57, 60, 66, 75, 78, 84, 87, 93, 96
I then compared them all together.
The numbers that were in two sets were:
21, 42, 45, 50, 55, 60, 75, 90
The numbers that were in three sets were:
10, 15, 30, 66, 78
There was no number that suited all sets.
The numbers that satisfied both Alison and Sam's statements went up in 20's starting with ten.