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Assume A, B, and C are positive integers. If AxB = C, then we say A
and B are factors of C.
Every positive integer greater than one has at least two factors : 1 and itself.
C
= B,
A
If
then we say A is a divisor of C when the quotient is an
integer. We can also say that C is divisible by A
Factor and Divisor these two words mean exactly the same thing.
Thus, we have three interchangeable way to say the same thing.
1) 8 is a factor of 24
2) 8 is a divisor of 24
3) 24 is divisible by 24
Similarly
1) 8 is not a factor of 12
2) 8 is not a divisor of 12
3) 12 is not divisible by 8
Because the quotient 12/8 is not an integer
If two number divide evenly
integer
= integer
integer
But, if two items don’t divide evenly, we have to possible options,
Option #1 : have an integer quotient and an integer remainder
8 goes into 12 once , with a remainder of 4
Option #2 : express quotient as a fraction or a decimal
𝟏𝟐
𝟖
=
𝟑
𝟐
=
𝟏
1𝟐
= 1.5
Divisibility Rules for 2
We have to do is look at the last digit. It must be even number.
For example 128, 264 = Yes, divisible by 2
Divisibility Rules for 5
We have to do is look at the last digit. It must be 5 or 0.
For example 105, 260 = Yes, divisible by 5
Divisibility Rules for 4
We have to do is look at the last two digit. It must be divisible by 4.
For example 196, 152 = Yes, divisible by 4
Divisibility Rules for 3
If the sum of the digit is divisible by 3, then the number is divisible by 3.
For example 102; 1+0+2 = 3, and since 3 is divisible by 3,
102 must be divisible by 3
Divisibility Rules for 9
If the sum of the digit is divisible by 9, then the number is divisible by 9.
For example 1296; 1+2+9+6 = 18, and since 18 is divisible by 9,
1296 must be divisible by 9
Divisibility Rules for 6
A number must be. (a) divisible by 2 (b) divisible by 3
For example 1296; (a) It’s eve
(b) Sum = 1+2+9+6 = 18, and since 18 is divisible by 3,
1296 must be divisible by 6
Multiple idea #1
 Just as 1 is a factor of every positive integer.
 Just as ever positive integer is a factor of itself.
Multiple idea #2
 If we need say the first five multiples of a number, we
simple multiply the original number by the number {1,2,3,4,5}
Multiple idea #3
 If P is a multiple of r, then (P - r) and (P + r) are also multiples
of r
Multiple idea #4
 If P and Q are multiples of r, then (P+Q) and (P-Q) must
also be multiples of r.
 If P is a multiple of r, then any multiple of P is a multiple of r
Multiple idea #6
 If P and Q are multiples of r, then the product P×Q must also
be a multiple of r.
 A prime number has only those two factor, 1 and itself.
 1 is NOT a prime number.
 2 is the only even prime number.
The Fundamental Theorem of Arithmetic
Every positive integer greater than 1 must either
(a) Be a prime number or
(b) Be expressed as a unique product of prime numbers.
The product is expressed as a product of prime number.
This product is called the prime factorization of the number
For examples.
10 = 2×5
12 = 3×4 = 3×2×2
24 = 8×3 = 2×2×2×3 = 𝟐𝟑 ×3
1.
2.
3.
4.
Find the prime factorization of N, and write it in terms
of powers of prime factors.
Create a list of the exponents of the prime factors.
Add one to every number on the list.
Find the product of the new list. That product is the
number of factor N has.
Squares of integers mean perfect square numbers
12 = 1x1
22 =
2x2
32 =
3x3
The exponents of the prime factors of a square most be even
N 2 = 𝑵 × 𝑵 = (𝒇𝒂𝒄𝒕𝒐𝒓 𝒐𝒇 𝑵)× (𝒇𝒂𝒄𝒕𝒐𝒓 𝒐𝒇 𝑵)
 Greatest Common Factor
 Greatest Common Divisor
 GCF
 In case of dealing number that are larger , we will use a
procedure involving the the prime factorizations
Find the GCE of 360 and 800
360 = 6 x 6 x 10 = 23x 32 x 5 = 23 x 5 x 32
800 = 8 x10 x 10 = 23x 5 = 23 x 5 x 22 x 5
Thus, GCF = 23 x 51 = 8 x 5 = 40
 Least Common Multiple (LCM)
 Least Common Denominator (LCD)
The LCM of 8 and 12
The multiples of 8:8,16,24,32,40,38,56,64,72,80,88,96,…
The multiples of 12 : 12,24,36,48,60,72,84,96,108,…
Least Common Multiple is 24
 The LCM is very important is adding and subtracting fractions ,
because the LCM is the LCD.
Relationship between the GCF and the LCM
For any two integers P and Q :
LCM =
𝐏×𝐐
𝐆𝐂𝐅
These are the odd numbers : …-7,-5,-3,-3,1,3,5,7,9…
These are the even numbers : …-8,-6,-4,-2,0,2,4,6,8,10…
• Even numbers can be expressed as 2k
• Odd numbers can be expresses in the from (2k+1) or (2k–1),
when k is any integers.
Adding & Subtracting Evens and Odds.
E+E=E
O+O=E
But. E + O = O
and
and
and
Multiplying Evens and Odds.
E×E =E
O×O =O
E×O =E
E-E=E
O-O=E
E-O=O
“consecutive” means “In a row ; one following another”
Some basic facts
1. A set of n consecutive integers will always contain one number
divisible by n
2. If n set is odd, then sum of a set of n consecutive integers will
always be divisible by n
3. In a set of 3 consecutive integers you could have two evens and
one odd, or two odds and one evens. In the set of 4 consecutive
integer you must have two evens and two odds.
when we divide one number by another number that is NOT one
of the factors
For examples.
20 ÷ 6 yields 3, with a remainder of 2
Terminology of divisor :
20 = the dividend
6 = the divisor
3 = the quotient
2 = the remainder
0 ≤remainder < divisor
If D = dividend, s = divisor ,Q = quotient, and r = remainder, we can
write:
𝐃
𝐫
=Q+
𝐒
𝐒
or
D = QS + r
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