We started by working out all the numbers up to 100 for each of the groups. Once we had these we checked to see if they were in each group.
The 2 digit numbers that belong in two sets are:
21,42,45, 55,70,75,90
The 2 digit numbers that belong in three sets are:
10, 15, 30, 60, 66, 78
When we got to work out the smallest number that fitted in each group we found we had to go further than 100. We figured that it would be the first triangular number that divides by 30 but not 90, because 30 fitted into 3 of the groups but not the triangular one. Therefore if the triangular number was divisible by 30 it would fit in all 4 groups. 210 was the first one we got to that did this.
When we looked at the relationship between set A and set S we found that if we looked at every number they had in common, each of these numbers divided by 10 would equal an odd number. The first numeral in the 2-digit number has to be odd, and the second numeral has to be 0 for this to work.
When we looked at the relationship between A and M, they contain every first and second multiple of 15, but not the third. The pattern restarts so 15, 30, 60,75, 105 etc. This is because the third, sixth, ninth etc multiple of 15 is divisible by 9.