Alison’s set- Multiples of five: 5, 10, 15, 20 , 25, 30, 35, 40, 45, 50, 55, 60…
It must end with 5 or 0.
Becky’s set- Triangular numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253…
Sam’s set- Even, but not multiples of 4: 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50…
Matt’s set- Multiples of 3, but not multiples of 9: 3, 6, 15, 21, 24, 30, 33, 39, 42, 48, 51, 57, 60…
Two digit numbers that belong to Alison and Becky’s set are 10, 15, 45, 55…
Two digit numbers that belong to Alison, Becky and Sam’s set is 10.
To find the smallest number that belongs in all four sets, we look into the triangular numbers that should be ended with 0, but not multiple of 4; multiple of 3 but not 9.
The triangular numbers that end with 0 are 10, 120, 210…
And the smallest number that satisfies all the rules is 210.
Numbers that belong to Alison and Sam’s sets are 10, 20, 50, 70, 90, 110…
The unit digit is always 0, and the tens digit is always odd.
Numbers that belong to Alison and Matt’s set: 15, 30, 60, 75, 105, 120…
Therefore, the numbers are multiples of 15 but not multiples of 9
Numbers that belong to Sam and Matt’s set: 6, 30, 42…
Therefore, the numbers are multiples of 6, but not multiples of 9 and 4.