MGF1106
Unit Two
Ex. 4.1 Number Theory
Objectives 18-20
Ex. 4.1
Objective 18
To write numbers in
prime factored form
Natural numbers greater than one can be
prime composite
classified as either ________ or _________.
Prime
_______ numbers are those natural greater
than 1 whose only factors are 1 and the
number.
Composite
___________ numbers are those natural
numbers greater than one that have factors
other than 1 and the number.
factor
A ________ is a multiplier.
The prime factorization of a number is when
the number has been written as a product of
prime numbers.
Write 540 in prime factored form.
540
22 ×33 × 5
10 54
2 5 6 9
2 3 3 3
Ex. 4.1
Objective 19
To find the number of
factors of a number
If n = paqbrc where p, q, and r are prime
numbers and a, b, and c are the number of
times each prime occurs in the number, the
the number of factors of n is found by
(a+1)(b+1)(c+1)
finding the product __________________.
Find the number of factors of 24 · 32.
(4+1)(2+1) = 5(3) = 15
Find the number of factors of 600.
Step 1: Write the number in prime factored
form.
600
10 60
2 5 6 10
2 3 2 5
23 ×31 × 52
600 = 23 ×31 × 52
Step 2: Find the number of factors.
(3+1)(1+1)(2+1) = 4(2)(3) = 24
Ex. 4.1
Objective 20
To find the sum of the
factors of a number
Find the sum of the factors of 24.
The sum of the factors of a number can be
found by using the prime factored form of the
number.
24 2 3
3 1
To do this, use the prime factors themselves.
Write the powers of each of the prime factors
beginning with 0 and going to the power of the
factor in the prime factored form.
0 1 2 3 0 1
2 ,2 ,2 ,2 3 ,3
The sum of these are formed for each of the
prime factors and then the product of these
sums in found.
(2 2 2 2 )(3 3 )
0 1 2 3 0 1
(1 2 4 8)(1 3)
(15)( 4)
60
Looking at this prime factored form of 600
which is 23 ×31 × 52, we would find the sum
by:
(20 + 21 + 22 + 23)(30 + 31)(50 + 51 + 52)
(1 + 2 + 4 + 8)(1 + 3)(1 + 5 + 25)
(15)(4)(31) = 1860
Find the sum of the factors of 31 · 23.
(30 + 31)(20 + 21 + 22 + 23)
(1 + 3)(1 + 2 + 4 + 8)
4(15) = 60