Chapter 4: Q55E (page 286)
Find the discrete logarithms of 5 and 6 to the base 2 modulo 19.
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Find the two’s complement representations, using bit strings of length six, of the following integers.
a) 22 b) 31 c) −7 d) −19
Describe a procedure for converting decimal (base 10) expansions of integers into hexadecimal expansions.
a) Define what it means for a and b to be congruent m odulo 7.
b) Which pairs of the integers-11,-8,-7,-1,0,3 and 17are congruent ?
c) Show that ifa and bare congruent m odulo 7, then 10a+13 and -4b+20 are also congruent m odulo 7.
It can be shown that every integer can be uniquely represented in the form
where, or 1 for j=0,1,2, …., k. Expansions of this type are called balanced ternary expansions. Find the balanced ternary expansions of
a) 5 .
b) 13 .
c) 37 .
d) 79 .
Show that ifis composite if and are integers greater than 1 and is odd. [ Hint: Show that is a factor of the polynomial if is odd.]
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