156462

First we can start with the numbers that satisfy Alison and Becky's statements but not Sam and Matt.
The numbers that satisfy Becky's statement follow this description n*(n+1)/2
To satisfy Alison and Becky's statement, either n or n+1 should be a multiple of five.
We get the following set of numbers: 10 15 45 55
10 satisfy Sam's statement, 15 satisfy Matt's statement. So the numbers that are left is 55 45.
We now do the same for Alison and Sam: 10 30 50 70 90
The numbers that is left will be: 50 70 90
Alison and Matt: 15 30 60 75
The numbers left: 60 75
Becky and Sam: 10 66 78
The numbers left: none
Becky and Matt: 15 21 66 78
The numbers left: 21
Sam and Matt: 30 42 66 78
The numbers left: 42
To conclude, the numbers that belong to EXACTLY two sets are: 21 42 45 50 55 60 70 75 90
The numbers that belong in three sets are the ones that are "kicked out" when calculating the numbers for exactly 2 sets, so the numbers in EXACTLY three sets are: 10 15 30 66 78
The smallest number that is in all 4 sets can be calculated. We can first work out the description of the set Alison intersection Sam intersection Matt.
The description: X divided by 180 have an remainder of 30 or 150
The set will be: 30 150 210 330 390 510 570 etc.
We try to find the smallest that satisfy n*(n+1)/2
We can find out the smallest is 210. The smallest number that satisfy all 4 statements is 210.