epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Introduction and Basic Terminology 33
Similarly, working backward through all the equaUties of the algorithm,
we obtain that r„ = GCD(r„_2, r„_3),..., r„ = GCD{a, b). •
26. If d= GCD{a, fo), then d = sa-{-tb for some integers s and t.
Proof: Using the steps of the Euclidean algorithm described in
Exercise 25, we obtain:
n = a — bqi
r2 = b ~ ri q2 = b - (a - bqi)q2 = as2 + bt2
r3 = n - r2 q3 = {a- bqi) - {as2 + ^^2)^3 = CIS3 + bt^.
Proceeding in this way, in at most b steps we will be able to write:
r„ = sa 4- tb.
The statement is therefore proved.
27. Let p be a prime number. If p divides the product ab, then p divides
either a or b.
•
Proof: If p does not divide a, then GCD(a, p) = i. (Explain why.)
Therefore,
1 = sa-{- pt
for some integers s and p. (See Exercise 26.)
Thus,
b = b(sa + pt)
= (kp)s + bpt
= p(ks + bt).
This implies that /? divides b. (Explain why.)
•
28. Let p be a prime number. Then, ^ is an irrational number.
Proof: Let us assume that ^ is a rational number; that is,
r- ^