Grade 6 Math Circles Worksheet - University Of Waterloo, 2015 Page 8
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10Theorems
Unique Factorization Theorem/Fundamental Theorem of Arithmetic
Any integer greater than 1 is either a prime number or can be written as a unique product
of prime numbers (ignoring the order).
Infinite Primes Theorem
There are infinitely many primes.
Proof:
Suppose that
= 2
= 3
are all of the primes. Let
= (
) + 1.
1
2
1
2
Let
be a prime dividing
. Then
can not be any of
,
, ...,
, because if were, the
1
2
remainder would be 1. So this prime
is still another prime, and
,
, ...,
would not be
1
2
all of the primes.
Wrap-Up
Today, you looked at prime numbers and their properties and uses. There are many more
theorems based on the prime numbers. They are a fascinating part of mathematics, and a
lot of research is still being done on this set of numbers.
Problem Set
Complete the following 15 problems. You may use a calculator.
1. Find the factors of the following numbers (use the divisibility tests when applicable)
(a) 96
(b) 4276
(c) 89
(d) 121
2. Find the prime factorization of the following numbers:
(a) 2459
(b) 456
(c) 123
(d) 54320
3. Mr. Batman wants to split his class of 24 students into equal groups for a group
project. List all of the possibilities of groups that Mr. Batman can divide his class
into so that no students are left without a group.
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