Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 331 — #339
9.13. References 331
Let
x WWD 17 88 31 5 37 2 59 1000
y WWD 19 .922/ 37 12 53 3678 59 29 :
(a) What is gcd.x; y/?
(b) What is lcm.x; y/? (“lcm” is least common multiple.)
Problem 9.5.
Show that there is an integer x such that
ax b .mod n/
iff
gcd.a; n/ j b:
Problem 9.6.
Prove that
for all a; b 2 Z.
gcd.a 5 ; b 5 / D gcd.a; b/ 5
Class Problems
Problem 9.7.
Use the Euclidean Algorithm to prove that
gcd.13a C 8b; 5a C 3b/ D gcd.a; b/:
Problem 9.8.
(a) Use the Pulverizer to find integers x; y such that
x30 C y22 D gcd.30; 22/:
(b) Now find integers x 0 ; y 0 with 0 y 0 < 30 such that
x 0 30 C y 0 22 D gcd.30; 22/