By David M. Burton
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Extra resources for Elementary Number Theory, Sixth Edition
But then whatever the choice of x and y, + cy = ar x + asy = a(r x + sy) Because rx + sy is an integer, this says that a I (bx + cy), as desired. 2 extends by induction to sums of more than two terms. That is, if a I bk fork = 1, 2, ... , n, then a I (b1x1 + bzxz + · · · + bnxn) for all integers x 1, x 2 , ••. , Xn. The few details needed for the proof are so straightforward that we omit them. If a and b are arbitrary integers, then an integer d is said to be a common divisor of a and b if both d I a and d I b.
Because 6 is the largest of these integers, it follows that gcd( -12, 30) = 6. In the same way, we can show that gcd( -5, 5) = 5 gcd(8 , 17) = 1 gcd( -8, -36) = 4 The next theorem indicates that gcd(a, b) can be represented as a linear combination of a and b. ) This is illustrated by, say, gcd( -12, 30) = 6 = (-12)2 + 30 · 1 or gcd( -8, -36) = 4 = (-8)4 + (-36)( -1) Now for the theorem. 3. Given integers a and b, not both of which are zero, there exist integers x and y such that gcd(a, b)= ax+ by Proof.
Inasmuch as a I c and b 1 c, integers rands can be found such that c = ar = bs. Now the relation gcd(a, b)= 1 allows us to write 1 =ax+ by for some choice of integers x andy. Multiplying the last equation by c, it appears that c = c · 1 = c(ax +by) = acx +bey If the appropriate substitutions are now made on the right-hand side, then c = a(bs)x + b(ar)y = ab(sx + ry) or, as a divisibility statement, ab I c. 24 ELEMENTARY NUMBER THEORY Our next result seems mild enough, but is of fundamental importance.