156546

Let n be the number of the data

(Alison) Multiples of 5:
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, etc. (5n)

(Becky) Triangular numbers:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153 etc. n(n+1)/2

(Sam) Even, but not multiples of four:
2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, etc. 4n-2

(Matt) “Multiples of three but not multiples of nine”
3, 6, 12, 15, 21, 24, 30, 33, 39, 42, 48, 51, 57, 60, 66, 69, 75, 78, 84, 87, 93, 96, 102, 105, 111, 114, 120, 123, 129, 132, 138, 141, 147, 150, 156, 159, 165, 168, 174, 177, 183, 186, 192, 195, 201, 204, 210, 213, 219, 222, etc.

Alison and Becky:
10, 15, 55, 105, 120, 190, 210, etc

Alison and Sam:
10, 30, 50, 70, 90, 110, 130, 150, 170, 190, 210, 230, 250 etc.

Alison and Matt:
30, 60, 120, 210, etc

Alison, Becky and Sam (Just look at the numbers above)
10, 210, etc

Alison Becky and Matt
120, etc

To find (the smallest number) out all of the sets you have to eliminate some of the numbers:

So, the four sets are
Multiples of 5
Triangular numbers
Even, but not multiples of four
Multiples of three but not multiples of nine

First, in the first set (multiple of 5) you have to cross out all of the multiples of fives that has a unit of 5 (e.g 15, 85, 145, etc) because one of the other sets says it’s an even number. So it leaves us with every other number i.e 20, 30, 40, 50, 60, etc. After that you have eliminate the numbers we’re left with which does not fit in both Alison’s and Matt’s Which then means we’re left with 30, 60, 120, 210. Then you have to eliminate those numbers which are no triangular numbers, so get rid of 30, 60, 120.

We are then left with 210.

For Alison and Sam:
10, 30, 50, 70, 90, etc
I found that to find the next number you had to add 20 to the number before because I found some when I laid out the numbers in the set and it started to make a pattern also you can find the answer by using the formula 10n+ 10(n-1)

For Alison and Matt:
30, 60, 120, 210, etc