Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 358 — #366
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Chapter 9
Number Theory
(k) If a b .mod .n// for a; b > 0, then c a c b .mod n/. true false
(l) If a b .mod nm/, then a b .mod n/. true false
(m) If gcd.m; n/ D 1, then
Œa b .mod m/ AND a b .mod n/ iff Œa b .mod mn/ true false
(n) If gcd.a; n/ D 1, then a n 1 1 .mod n/ true false
(o) If a; b > 1, then
[a has a inverse mod b iff b has an inverse mod a]. true false
Problem 9.69.
Find an integer k > 1 such that n and n k agree in their last three digits whenever n
is divisible by neither 2 nor 5. Hint: Euler’s theorem.
Problem 9.70.
(a) Explain why . 12/ 482 has a multiplicative inverse modulo 175.
(b) What is the value of .175/, where is Euler’s function?
(c) Call a number from 0 to 174 powerful iff some positive power of the number
is congruent to 1 modulo 175. What is the probability that a random number from
0 to 174 is powerful?
(d) What is the remainder of . 12/ 482 divided by 175?
Problem 9.71. (a) Calculate the remainder of 35 86 divided by 29.
(b) Part (a) implies that the remainder of 35 86 divided by 29 is not equal to 1. So
there there must be a mistake in the following proof, where all the congruences are