Combinatorics Through Guided Discovery, 2004a

150 B. Mathematical Induction
cent stamps, so with one more five cent stamp, so can k cents. Thus by the (strong)
principle of mathematical induction, we can make n cents in stamps with five and
six cent stamps for each n ≥ 20.
Some people might say that we really had five base cases, n =20, 21, 22, 23, and
24, in the proof above and once we had proved those five consecutive base cases,
then we could reduce any other case to one of these base cases by successively
subtracting 5. That is an appropriate way to look at the proof. A logician would
say that it is also the case that, for example, by proving we could make 22 cents, we
also proved that if we can make 20 cents and 21 cents in stamps, then we could also
make 22 cents. We just didn’t bother to use the assumption that we could make 20
cents and 21 cents! So long as one point of view or the other satisfies you, you are
ready to use this kind of argument in proofs.
Problem 368. A number greater than one is called prime if it has no factors
other than itself and one. Show that each positive number is either a prime
or a power of a prime or a product of powers of prime numbers.
Problem 369. Show that the number of prime factors of a positive number
n ≥ 2 is less than or equal to 2 n. (If a prime occurs to the kth power in a
factorization of n, you can consider that power as k prime factors.) (There
is a way to do this by induction and a way to do it without induction. It
would be ideal to find both ways.)
Problem 370. One of the most powerful statements in elementary number
theory is Euclid’s Division Theorem. a This states that if m and n are positive
integers, then there are unique nonnegative intergers q and r with 0 ≤ r < n,
such that m = nq + r. The number q is called the quotient and the number
r is called the remainder. In computer science it is common to denote r by
m n. In elementary school you learned how to use long division to
find q and r. However, it is unlikely that anyone ever proved for you that
for any pair of positive intgers, m and n, there is such a pair of nonnegative
numbers q and r. You now have the tools needed to prove this. Do so. (h)
a In a curious twist of language, mathematicians have long called The Division Algorithm or
Euclid’s Division Algorithm. However as computer science has grown in importance, the word
algorithm has gotten a more precise definition: an algorithm is now a method to do something.
There is a method (in fact there are more than one) to get the q and r that Euclid’s Division
Theorem gives us, and computer scientists would call these methods algorithms. Your author
has chosen to break with mathematical tradition and restrict his use of the word algorithm to
the more precise interpretation as a computer scientist probably would. We aren’t giving a
method here, so this is why the name used here is “Euclid’s Division Theorem.”