Why Read This Book? - Index of

2.5 Divisibility and Prime Numbers 61
(D1) g | a and g | b.
(D2) If h is any positive integer such that h | a and h | b, then h | g also.
Proof. Let a and b be nonzero integers, and define
S ={ma + nb : m, n ∈ Z,ma+ nb > 0} (2.16)
That is, S is the set of all positive integer linear combinations of a and b. First, S is
not empty, for depending on the signs of a and b, we may let m, n =±1 to produce
some positive value of ma + nb. By the WOP, S contains a smallest element g,
which may be written in the form g = m0a + n0b for some integers m0 and n0.By
Exercise 2.5.8, g has properties D1–D2, and if g1 and g2 are both positive integers
that have properties D1–D2, then g1 = g2.
EXERCISE 2.5.8 Let g = m0a + n0b be the smallest element of S as defined in
Eq. (2.16). Show the following to complete the proof of Theorem 2.5.7.
(a) g | a. (The proof that g | b is identical.) 18,19
(b) If h is any positive integer with the properties that h | a and h | b, then it
must be that h | g.
(c) If g1 and g2 are positive integers with properties D1–D2, then g1 = g2. 20
The proof of Theorem 2.5.7 yields a serendipity about gcd(a, b), in that it can
be written as an integer linear combination of a and b, and the smallest such positive
expression is in fact the gcd. This can be particularly helpful if gcd(a, b) = 1,
in which case we say that a and b are relatively prime. Immediately, we can see
that if a and b are relatively prime, then there exist integers m and n such that
ma + nb = 1. Furthermore, if it is possible to find an integer linear combination
of a and b that equals 1, then clearly this linear combination is the smallest such
positive linear combination. Thus a and b are relatively prime.
EXERCISE 2.5.9 Show that 14 and 33 are relatively prime by determining
integers m and n such that 14m + 33n = 1.
Now we are ready to define prime numbers and investigate a few of their
important properties. A positive integer p is defined to be prime if it has precisely
two distinct positive integer divisors. If this is true, then these divisors must be
1 and p. Note that this definition excludes 1 from being prime.
Theorem 2.5.10 Suppose a is a nonzero integer and p is prime. Then either a
and p are relatively prime or p | a.
18 Apply the division algorithm to a and g and show that r>0 is impossible.
19 To suppose r>0 leads to a contradiction because it produces an element of S that is smaller than g.
20 Theorem 2.5.5 should come in handy.