Chapter 4: Q58E (page 286)
Which integers are quadratic residues of 11?s
The quadratic residues of 11 are 1,3,4,5,9
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It can be shown that every integer can be uniquely represented in the form
where, or 1 for j=0,1,2, โฆ., k. Expansions of this type are called balanced ternary expansions. Find the balanced ternary expansions of
a) 5 .
b) 13 .
c) 37 .
d) 79 .
Describe a procedure for converting decimal (base 10) expansions of integers into hexadecimal expansions.
Prove that for every positive integer, there are consecutive composite integers. [ Hint: Consider the consecutive integers starting with ].
Convert to octal and hexadecimal representations.
The value of the Euler -function at the positive integer is defined to be the number of positive integers less than or equal to that are relatively prime to. [Note: is the Greek letter phi.]
Find these values of the Euler -function.
a)role="math" localid="1668504243797" b)role="math" localid="1668504251452" c)role="math" localid="1668504258881"
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