Math 412. Worksheet on The Euclidean Algorithm.
Professors Jack Jeffries and Karen E. Smith
DEFI NI TI ON : The greatest common divisor or GCD of two integers a, b is the largest integer
dsuch that d|aand d|b. We often write (a, b)for the GCD of aand b.
THE OR EM 1.2: Let aand bbe integers, and assume that aand bare not both zero. There exist
r, s ∈Zsuch that ra +sb = (a, b).
The Euclidean algorithm is a method to find the GCD of two integers, as well as a specific
pair of numbers r, s such that ra +sb = (a, b). We will say that an expression of the form
ra +sb with r, s ∈Zis a linear combination of aand b.
A. WARMU P:
(1) List all factors1of 18? List all factors of 24. Find (18,24).
(2) For a∈Z, what is (a, a)? What is (a, 7a)? If a > 0, what is the GCD of aand 0?
ANSWER:
(1) The factors of 18 are {±1,±2,±3,±6,±9,±18}.
The factors of 24 are {±1,±2,±3,±4,±6,±8,±12,±24}. The GCD is 6.
(2) (a, a) = |a|,(a, 7a) = |a|and (a, 0) = |a|.
B. Suppose we had two numbers aand b, and we did the division algorithm to get a=bq +r
for some q, r ∈Z.
(1) Show that if dis a common divisor of band r, then dis a common divisor of aand b.
What does this say about the relationship between (a, b)and (b, r)?
(2) Show that if dis a common divisor of aand b, then dis a common divisor of band r.
What does this say about the relationship between (b, r)and (a, b)?
(3) Show that (a, b)=(b, r).
(4) How might (3) make the computation of (a, b)easier?
ANSWER:
(1) Let dbe a common factor of band r. So there exist k1, k2∈Zsuch that b=dk1and r=dk2.
Substitute a=bq +r=dk1q+dk2=d(k1q+k2). Since k1q+k2∈Z, we have d|a. Thus dis
a common factor of aand b. We conclude that (b, r)≤(a, b)since the greatest common factor of b
and ris at least some common factor of aand b, but not necessarily the greatest.
(2) Let dbe a common factor of band a. So there exist k1, k2∈Zsuch that b=dk1and ra =dk2.
Substitute r=a−bq =dk1−dk2q=d(k1−k2q). So similarly, d|r. Thus dis a common factor
of rand b. By a similar argument, we conclude that (b, r)≥(a, b)
(3) This follows from (1) and (2), since both (b, r)≥(a, b)and (b, r)≤(a, b).
(4) The integers aand bmay have an out-of-control number of digits, but assuming (without loss of
generality) that a>b, then we know that the pair b, r will be “smaller”. So instead we can compute
the GCD (b, r), which is the same as the one we started with (a, b).
1Factor is another word for divisor. Completely synonymous.
