Let a, b (= N (so, a not= 0) and x, r (= Z. Using only the definition of divisibility along with the

Axioms of Integers

If a | b and a | (bx + r), then a | r.

According to the definition, a | b means that b = a*p for some integer p. Similarly, a | (bx + r) implies that b*x + r = a*q for some integer q.

Next, r = (b*x + r) - b*x = a*q - a*p*x - use associative and commutative axioms of addition and multiplication for integers.

Then r = a(q - p*x) - use distributive axiom.

Finally, by the definition of divisibility, a | r.

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