epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

126 The Nuts and Bolts of Proof, Third Edition
19. By hypothesis (a H- b)^ = a^ -\-b^. But, by the rules of algebra
{a + hf = a^ -f lab + b^. Thus, we obtain that, for all real numbers
b, a^-{•2ab-\-b^ = a^-\-b^. This implies that 2ab = 0 for all real
numbers b. In particular, this equaUty is true when b ^ 0. Thus, we
can divide the equahty lab = 0 by 2b and obtain a = 0.
20. (Because it is impossible to check directly all prime numbers that can
be written in the form 2" — 1 to see if the corresponding exponent n is
indeed a prime number, we will try to use the contrapositive of the
original statement.) We will assume that n is not a prime number.
Then n is divisible by at least another number t different from n and 1.
So, t>l. Thus, n = tq where q is some positive integer, ^ / 0 (because
n^O), q^\ (because t^^^n), and ^7«^/i (because t^ 1). Therefore, we
can write:
2«-l = 2'^-l = (2^y-l.
We can now use factorization techniques to obtain:
2" - 1 = (2^y-l = {T - l)[(2^y-V(2^)'-V ••• + !].
This equality shows that 2" — 1 is not a prime number because it can
be written as the product of two numbers, and neither one of these
numbers is 1. (Why? Look at all the information hsted above regarding
t and q) Because we proved that the contrapositive of the original
statement is true, we can conclude that the original statement is true
and n must be a prime number.
21. Let n be a four-digit palindrome number. To prove that it is divisible
by 11, we need to show that n=llt for some positive integer t.
Because n is a four-digit palindrome number, we can write n — xyyx,
where x and y are integer numbers between 0 and 9, and x^Q. We
can now try to separate the digits of n. Thus,
Because t = 9lx-^l0y
divisible by 11.
n = xyyx = lOOOx + lOOj; + lOy + x
= lOOlx 4-110}; = ll(91x + lOy).
is a positive integer, we proved that n is
22. We know that x^c. So, the denominator of the fraction is a nonzero
number, and the fraction is well defined. We want to prove that the
number
m-f(x)