Math Circles Prime Time Worksheet With Answers - Grade 6 - 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
1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
(b) 4276
1, 2, 4, 1069, 2138, 4276
(c) 89
1, 89
(d) 121
1, 11, 121
2. Find the prime factorization of the following numbers:
(a) 2459
2459
(b) 456
2
2
2
3
19
(c) 123
3
41
(d) 54320
2
2
2
2
5
7
97
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