Math 412. Worksheet on The Euclidean Algorithm.
Professor Karen E. Smith
DEFI NI TI ON : Let aand bbe integers, not both zero. The greatest common divisor or gcd of
a, b is the largest integer dsuch that d|aand d|b. We write gcd(a, b), or sometimes just (a, b),
for the gcd of aand b.
THE OR EM : Let aand bbe integers, not both zero. Then there exist (non-unique!) r, s ∈Zsuch
that ra +sb =gcd(a, b).
DEFI NI TI ON : An expression of the form ra +sb with r, s ∈Zis a called a Z-linear combina-
tion of aand b.
The Euclidean algorithm is a method to find the gcd of two integers, as well as a specific pair
of integers r, s such that ra +sb =gcd(a, b).
A. WARMU P:
(1) List all factors of 18? List all divisors of 24. Find gcd(18,24).
(2) For a∈Z, what is (a, a)? What is (a, 7a)? If a > 0, what is the gcd of aand 0?
(3) What happens if we try to define gcd(0,0)?
B. Suppose we had two integers aand b, with b > 0, and we did the division algorithm to get
a=bq +rfor some q, r ∈Z.
(1) Show that if dis the greatest common divisor of band r, then dis acommon divisor of
aand b. Now explain why this means that (b, r)≤(a, b).
(2) Show that if dis the greatest common divisor of aand b, then dis acommon divisor of
band r. Now explain why this means that (a, b)≤(b, r).
(3) Prove that gcd(a, b) = gcd(b, r).
(4) How might (3) make the computation of (a, b)easier?
C. Your team is tasked with computing the greatest common divisor of 524 and 148. Your
teammate does the following computation, which you can assume is accurate:
524 = 148 ·3+80 0≤80 <148(i)
148 =80 ·1+68 0≤68 <80(ii)
80 =68 ·1+12 0≤12 <68(iii)
68 =12 ·5+80≤8<12(iv)
12 =8·1 + 4 0 ≤4<8(v)
8= 4 ·2+0(vi)
(1) What is going on on each individual line?
(2) How does each line relate to the previous one?
(3) Prove that
(524,148) = (148,80) = (80,68) = (68,12) = (12,8) = (8,4) = (4,0) = 4.
D. The computation above is an example of the Euclidean algorithm applied to 524 and 148.
(1) Articulate with your teammates, perhaps by drawing a flow-chart, how the Euclidean
algorithm is carried out to compute the gcd of two positive integers aand b.
(2) Use the Euclidean algorithm to find (1003,456).
