 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-mathematics-continuous-mathematics-and-probability/3/b/ii/solution

a = q b + r

with

q \in N_{0}

and

0 \leq r < b

a = n u, b = n v

gives

r = a - q b = n (u - q v) .

Since

u - q v

is an integer, this proves the divisibility

n ∣ r,

including

r = 0

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