Rounding during integer division, how to understand?

Question

Algorithms

Task

1476A - Sum multiple of K

https://codeforces.com/problemset/problem/1476/A

Problem analysis

here

https://codeforces.com/blog/entry/87356

Incomprehensible moment:
It is quite obvious that the smaller s, the smaller the maximum ai, that is, we need to find the minimum cf such that cf⋅k≥n. Then cf=⌈n/k⌉=⌊(n+k−1)/k⌋.

How did ⌈n/k⌉ become ⌊(n+k−1)/k⌋ ?

Answer 1

Let n = pk + d, where 1 <= d <= p
Then ⌈n/k⌉ = p + 1
n + k - 1 = pk + d + k - 1 = (p + 1)k + (d - 1 )
⌊(n+k−1)/k⌋ = ⌊p + 1 + (d - 1)/k⌋ = p + 1, because 0 <= d - 1 < k