I assume by smallest number, you're really looking for the smallest *positive* number.
Notice that -5 fits all these requirements:
[tex]-5\equiv15\mod20[/tex]
[tex]-5\equiv20\mod25[/tex]
[tex]-5\equiv25\mod30[/tex]
[tex]-5\equiv31\mod36[/tex]
[tex]-5\equiv43\mod48[/tex]
Any number of the form [tex]-5+nk[/tex] will also satisfy these conditions, where [tex]k\in\mathbb Z[/tex] and [tex]n[/tex] is the least common multiple of the moduli. You have
[tex]\mathrm{lcm}(20,25,30,36,48)=3600[/tex]
so the least positive number would be achieved with [tex]k=1[/tex], giving 3595 as the answer. (Verified with a script.)
