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Blast into Math! Prime nummers: indestructimle muilding mlocks

Corollary 5.1.5 (Chocolate Chip Corollary (CCC)). Let c ∈ Z , c =0, and k ∈ N . If p is prime, and
p|c , then p|c .
k

Proof: To prove this corollary, we’ll use the CCP: that’s why we can call it the Chocolate Chip Corollary.

The CCP tells us that if a prime number divides a product of a finite sequence of non-zero integers,
then that prime divides at least one of those integers. In the corollary, the prime number p divides c
k
which is c times itself k times. So, the prime number divides the product of the elements of the finite
sequence S = {c, c,...,c} , which is the finite sequence consisting of the integer c repeated k times.
By the CCP, since p divides the product of the elements of S, p divides at least one element in S. But,
every element of s is equal to c, which means that p|c .

♥
5.2 Unique prime factorization: the Fundamental Theorem of Arithmetic

You may have noticed that we use many abbreviations, like the IP, SP, CCP, CCC, and the FTA. You have

also seen how two of the most important numbers, the multiplicative and additive identities disguise
themselves and act as spies in equations. Mathematicians have something like a secret society, because we
spend so much of our time working with mathematical concepts that are only known and understood by
other mathematicians who work on the same concepts. These concepts have names, and when the names
are long, we often abbreviate them. When we talk about these concepts, it’s like we are speaking a secret
mathematical language. When we write about mathematics, we also use secret mathematical symbols like
∈ , ∀, and ∅ . Let’s use the secret language we have created together in this chapter to prove the FTA!

Exercise: Make a list of the mathematical symbols and abbreviations you have learned in this book so far.

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