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Solution. Deﬁne I = {k in Z : ζ k = 1}. Of course, I is a subset of Z, and
it contains positive numbers because ζ is a root of unity. Observe that
(i) If k is in I and u is in Z, then uk is in I :
If k is in I , then ζ k = 1. Hence, ζ uk = (ζ k )u = 1u = 1, and so
uk is in I .
(ii) If k , are in I , then k + is in I :
If k , are in I , then ζ k = 1 = ζ , so that 1 = ζ k ζ
hence, k + is in I . = ζ k+ ; It now follows from Corollary 1.37 that there is a positive number d in I ,
i.e., ζ d = 1, such that every k in I is a multiple of d ; that is, if ζ k = 1,
then d  k .
1.52 Show that every positive integer m can be written as a sum of distinct powers of 2; show, moreover, that there is only one way in which m can so be
written.
Solution. In base 2, the only digits are 0 and 1. If we neglect the binary
digits 0, then every positive integer is uniquely a sum of powers of 2.
1.53 Find the badic digits of 1000 for b = 2, 3, 4, 5, and 20.
Solution.
base 2 :
base 3 :
base 4 :
base 5 :
base 20 : 1000 = 1111101000
1000 = 1101001
1000 = 33220
1000 = 13000
1000 = 2θ, where θ is a symbol denoting the new digit “ten.”
1.54
(i) Prove that if n is squarefree (i.e., n > 1 and n is not divisible by
√
the square of any prime), then n is irrational.
Solution. We rewrite the proof of Proposition 1.14. Suppose, on
√
the
√ contrary, that n is rational, where n is squarefree; that is,
n = a /b. We may assume that a /b is in lowest terms; that is,
(a , b) = 1. Squaring, a 2 = nb2 . Let p be a prime divisor of n ,
so that n = pq . Since n is squarefree, ( p , q ) = 1. By Euclid’s
lemma, p  a , so that a = pm , hence p 2 m 2 = a 2 = pqb2 , and
pm 2 = qb2 . By Euclid’s lemma, p  b, contradicting (a , b) = 1.
√
(ii) Prove that 3 2 is irrational.
√
3
Solution. Assume that 4 = a /b, where (a , b) = 1. Then 4b3 =
a 3 , so that a is even; say, a = 2m . Hence 4b3 = 8m 3 ; canceling,
b3 = 2m 3 , forcing b to be even. This contradicts (a , b) = 1. ...
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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