Question

which smallest number will have 15,20,25,31 nd 43 as reminder respectively when divided by 20, 25, 30, 36,48
please explain the answer you found

Answer 1

I assume by smallest number, you're really looking for the smallest *positive* number.

Notice that -5 fits all these requirements:

$-5\equiv15\mod20$
$-5\equiv20\mod25$
$-5\equiv25\mod30$
$-5\equiv31\mod36$
$-5\equiv43\mod48$

Any number of the form $-5+nk$ will also satisfy these conditions, where $k\in\mathbb Z$ and $n$ is the least common multiple of the moduli. You have

$\mathrm{lcm}(20,25,30,36,48)=3600$

so the least positive number would be achieved with $k=1$, giving 3595 as the answer. (Verified with a script.)