For ex., say a number 120016, if I need to find if the number is divisible by 16 (=2^4), I just need to check if the last 4 digits is divisible by 4. So here 120016 is divisible by 16, because the last 4 digit is divisible by 16.

Now, let’s not convinced just with some shortcut.

Let’s understand the concept by taking a 5 digit number represented by abcde, where a,b,c,d,e each represents some decimal from 0 to 9.

I need to find out if the number abcde is divisible by 8 (=2^3).

Expressing the number abcde as ab000 +cde

So now, ab000 is nothing but abX1000, no doubt this is divisible by 8. Because 1000 is a factor for it which is multiple of 8.

Now we just need to confirm if the number cde is divisible by 8. The same applies for any number X , to confirm if it is divisible by 2^n we just need to check if the last n digits is divisible by n.

The similar way we can prove for powers of 5!