First of all, the number has to be a multiple of 5! So, it has to be 5n!
Second of all, the number has to have a factor of 2, but not 4! We can write that as 2n( n is not a even number)!
Third of all, it has to be have a factor of 3, but not 9! We can write that as 3n(n can't be divided by 3)
This number has to sutible for all three patterns. We have to caculate the smallest common mutiple of those 3 numbers, which is 30n(n is not a even number, n can't be divided by 3)
There we look at the triangler number part, because of the number 30, there is only one possibility that can be sutible for these 4 patterns. That is the units digit is 0.
The triangler number formula is n(n+1)/2, this mean that the n's unit has to be 0. Now we just try!
20*(20+1)/2= 210
210/3=70 210/9=23.....7 210/5=42 210/2=105 210/4=52......2