A number when divided successively by 4 and 5 leaves

The question means something different from what it actually says, but I can guess what it's trying to say.

When we take our number, "n", and divide it by 4, we get a remainder of 1. So n is exactly 1 greater than some multiple of 4, and we have:

n = 4q + 1

It's here where the wording is confused. We are not now "successively" dividing our number n by 5, so we are not dividing n/4 by 5 (which wouldn't make sense in a remainders question, because n/4 is not an integer at all). Instead we're meant here only to be dividing the resulting quotient by 5, ignoring the remainder. So it's only q in the equation above that we're dividing by 5. Since we get a remainder of 4 when we divide q by 5, and we have:

q = 5k + 4

And now we can just plug that in for q in our first equation above:

n = 4q + 1 = 4(5k + 4) + 1 = 20k + 17

and we can see that n is 17 larger than some multiple of 20, and thus when we divide it by 20, the remainder will be 17.