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-3m -2m -m 0 m 2m 3m 4m
First note that a set S of numbers is "closed under subtraction" if "p is in S" and "q is is S" imply that p-q is in S. Here's a nice statement stolen from Andre Weil's "Number Theory for Beginners":
Theorem. If a non-empty subset M of the integers is closed under subtraction, then there is a unique natural number m such that M is the m-times-table.Proof. Since M is non-empty, it has an element x. Then 0 is in M since x-x is. Then -x is in M since 0-x is. Then if y and z are in M then y+z is, since y-(-z) is (so M is closed under addition). Using this, if y is in M then ny is in M by induction, for positive integers n, and -ny is in M as before.
First, M={0} is OK: just take m=0. Otherwise M contains positive integers. By the Least Integer Axiom, let m be the smallest one. All multiples of m are in M, as we showed last paragraph. Now arguing by contradiction, if y were an element of M which is NOT a multiple of m then y = qm +r with 0 < r < m. But r=y-qm is in M and this contradicts the smallestness of m. So the existence of an element y of M which is not a multiple of m is impossible. qed
Corollary 1. If {a,b,...,z} is any finite set of integers, and D is the set of all linear combinations r1a+r2b+...+rjz with arbitrary integer coefficients ri, then D is the d-times-table for a natural number d.
Proof. Just check that the difference of two elements of D is also in D, then use the Theorem.
Corollary 2. The number d in Corollary 1 is the gcd of {a,b,...,z}.
Proof. Since D is the d-times-table and the individual numbers a,b,...,z are all elements of D, d divides each of them. Any other common divisor of {a,b,...,z} divides any linear combination of them, in particular d. qed
What just happened is a good example of the benefits of abstraction. The statement that gcd(a,b)=sa+tb for some integers s and t is a corollary of the Corollaries! Along the way, we proved that any subset of the integers which is closed under subtraction consists of exactly the multiples of one positive integer. So the set of ALL possible sa+tb's (for fixed a and b) "looks" like a times-table.