The smallest number that can be used for all four statements (multiple of five, multiple of two but not four, multiple of three but not nine, and triangular numbers) is 210. I found this out by looking at Alison's, Matt's and Sam's statements (multiple of five, multiple of two but not four, and multiple of three but not nine) first.
The sequence for multiples of five is 5, 10, 15, 20, 25, 30, 35, 40, 45...
The sequence for multiples of two but not four is 2, 6, 10, 14, 18, 22, 26, 30, 34...
The sequence for multiples of three but not nine is 3, 6, 12, 15, 21, 24, 30, 33, 39...
A triangular number is a number that follows the sequence 1, 3, 6, 10, 15, 21, 28, 36, 45, 55... This sequence is created by the simple pattern +1, +2, +3, +4, +5...
The easiest way to find this out is to find the sequence that follows the Alison's, Matt's and Sam's rules. I figured this out by writing all the multiples of five that can be divided by two but not four. I ended up with a list like this: 10, 30, 50, 70, 90, 110, 130, 150, 170, 190, 210, 230, 250... As you can see it is a pattern that starts with 10 and increases by 20. Out of all these numbers I eliminated the numbers that could be divided by 9 and could not be divided by 3. My sequence became 30, 150, 210, 330, 390, 510, 570... As you can see this is a pattern that starts with 30 and increases by 120 and 60.
Out of these numbers I eliminated those that were not a triangular number. I was left with only 210 which was my final answer. This can be proven by checking if 210 can be placed into all categories.
Number needs to be divided by 5 210 / 5 = 24
Number needs to be divided by 2 210 / 2 = 105
Number cannot be divided by 4 210 / 4 = 52.5
Number needs to be divided by 3 210 / 3 = 70
Number cannot be divided by 9 210 / 9 = 23.33
Number needs to be triangular 1 + 2 + 3 + 4... ... + 17 + 18 + 19 + 20 = 210
Answers to Alison's and Becky's statements: 10, 15, 45, 55
Answers to Alison's and Sam's statements: 10, 30, 50, 70, 90
Answers to Alison's and Matt's statements: 15, 30, 60, 75
Answers to Becky's and Sam's statements: 6, 10, 66, 78
Answers to Becky's and Matt's statements: 3, 6, 15, 21, 66, 78
Answers to Sam's and Matt's statements: 6, 30, 42, 66, 78
Answer to Alison's, Becky's and Sam's statements: 10
Answer to Alison's, Becky's and Matt's statements: 15
Answer to Alison's, Sam's and Matt's statements: 30
Answers to Becky's, Sam's and Matt's statements: 6, 66, 78
I noticed with the numbers that satisfy Alison's and Sam's is that is starts with 10 and increases by 20.
I noticed with the numbers that satisfy Alison's and Matt's statements is that it starts with 15 and increases by 15.
Thanks and please publish my solution.
-Gabriel Soto
P.S. Awesome website.