 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-resit/3/b/i/solution

a_{1} - b_{1} = n u

and

a_{2} - b_{2} = n v

for some integers

u, v

. Adding gives

(a_{1} + a_{2}) - (b_{1} + b_{2}) = n (u + v) .

Since

u + v

is an integer, this proves

a_{1} + a_{2} \equiv b_{1} + b_{2} (mod n) .

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