Chapter 8 Number Theory302
Problem 8.52.
(a)What is the probability that an integer from 1 to 360 selected with uniform
probability is relatively prime to 360?
(b)What is the value of rem.7^98 ; 360/?Class Problems
Problem 8.53.
Find the remainder of 261818181 divided by 297.
Hint: 1818181 D.18010101/C 1 ; use Euler’s theorem.
Problem 8.54.
Find the last digit of 77
77
.
Problem 8.55.
Prove thatnandn^5 have the same last digit. For example:
25 D 32 795 D 3077056399Problem 8.56.
Use Fermat’s theorem to find the inverse,i, of 13 modulo 23 with 1 i < 23.
Problem 8.57.
Let be Euler’s function.
(a)What is the value of .2/?(b)What are three nonnegative integersk > 1such that .k/D 2?(c)Prove that .k/is even fork > 2.Hint:Consider whetherkhas an odd prime factor or not.
(d)Briefly explain why .k/D 2 for exactly three values ofk.