123
(f) 43 mod 9 = 7
43 29 (mod 9).
29 mod 9 = 2
18. The statement is false. Let maxcfw_a, b, c = c. Then 0 < |a b| < c and c
a b (mod c).
(a b). So
Exercises for Section 7.5. Introduction to Cryptogr
120
21. Proof. Let p 5 be a prime. Dividing p by 6, we obtain p = 6k + r for some integers
k and r, where 0 r 5. If r = 0, then 6 | p, which is impossible. If r = 2, then
p = 6k + 2 = 2(3k + 1). Since
115
11. Proof. If n = 1, then n3 + 1 = 2 is a prime. Assume, to the contrary, that there exists a prime
p = n3 + 1 for some integer n 2. Then p = (n + 1)(n2 n + 1). Therefore, either n + 1 = 1
or n2 n
76
(b) The distinct equivalence classes resulting from R are S1 , S2 , , Sk .
16. Observe that
(1) tells us that there are three equivalence classes.
(2) tells us that no equivalence class has exactly
73
(d) The relation R4 is symmetric but is neither reexive nor transitive. For example, (1)
|cfw_1, 2cfw_1, 2| = 2 and (2) |cfw_1, 2cfw_1| = 1 and |cfw_1cfw_1, 2| = 1 but |cfw_1, 2cfw_1, 2| = 2.
Exerc
58
By the Principle of Mathematical Induction,
Fn1 Fn+2 = Fn Fn+1 + (1)n
for every integer n 2.
Section 4.4. The Strong Principle of Mathematical Induction
1. (a) a2 = 2, a3 = 4, a4 = 8 and a5 = 16.
(
127
Hence gcd(ak+1 , ak+2 ) = 1.
By the Strong Principle of Mathematical Induction, gcd(an , an+1 ) = 1 for every positive integer
n.
26. Proof. Assume, to the contrary, that log2 3 is rational. Then