Unit 1: Number Theory
Word
Definition
rectangular array
a rectangular arrangement of objects in rows and columns such that each row has
the same number of objects and each column has the same number of objects.
number model
a number sentence that models or fits a situation
Example: Harry is 8 years older than his 5 year old sister can be modeled by the
number sentence 8 + 5 = 13.
turn-around rule
o
o
o
factor
factor pair
a rule for solving addition and multiplication problems based on the
commutative property
Example: 6 X 8 =48, the turnaround rule is 8 X 6 = 48
Commutative property- the changing of the order of the numbers being
added or multiplied doesn't change the answer.
one of two numbers that are multiplied to give a product
o
o
two whole-number factors of a number
a number may have more than one factor pair
Example: the factor pairs for 18 are 1 and 18, 2 and 9, and 3 and 6
factor rainbow
a way to show factor pairs in a list of all the factors of a number
product
the result of multiplying two numbers
divisible by
one whole number is divisible by another whole number if there is not remainder
divisibility test
a test to find out whether a whole number is divisible by another whole number
without actually doing the division.
quotient
the result of dividing one number by another number
Example: In 35/5 =7, the quotient is 7.
remainder
an amount left over when one number is divided by another number.
Example: 35 divided by 5 = 7 equal groups with a remainder of 3
composite
number
a whole number that has more than two factors
prime number
a whole number that has exactly two factors, itself and 1.
square array
is a special rectangular array that has the same number of rows as it has columns
square number
a number that is a product of a whole number multiplied by itself
exponent
a small raised number in exponential notation that tells how many times the base
is to be multiplied by itself
base
the number that is raised to some power
exponential
notation
a way to show repeated multiplication by the same factor
3
Example: 2 is exponential notation for 2 X 2 X 2
square root
4 is the square root of 16 because 4 X 4 = 16
name-collection
box
a box-like diagram tagged with a given number and used for collecting equivalent
names for that number.
factor string
a number written as a product of at least two whole-number factors.
Example: a factor string for the number 24 is 2 X 3 X 4. This factor string has
three factors, so its length is 3. The number 1 is never part of a factor string.
factor tree
a way to get the prime factorization of a number. The original number is written as
a product of factors; then each of these factors is written as a product of factors
and so on, until the factors are all prime numbers.
prime
factorization
a whole number expressed as a product of prime factors.
Example: the prime factorization of 24 is 2 X 2 X 2 X 3.
Factors
o
o
o
are a way to breakdown numbers
help you understand the relationships between numbers
there are different ways to express factors
Prime Factors of the number 12
12 = 2 X 2 X 3 the numbers 2 & 3 are prime numbers
Factors of the number 12
1 & 12, 2 & 6, 3 & 4
Factor Rainbow for the number 12
1, 2, 3, 4, 6, 12
Divisibility Rules
Help you decide if a number will be divisible by another number. If you know these tests it can
help make dividing easier for you.
A number is divisible by 2- if it is an even number
A number is divisible by 3- if the sum of the digits is divisible by 3.
(534 is divisible by 3 because 5 + 3 + 4=12 and 12 is divisible by 3)
A number is divisible by 6-if it is divisible by 2 & 3.
A number is divisible by 9-if the sum of the digits is divisible by 9.
(8,253 is divisible by nine because 8 + 2 + 5 + 3= 18)
A number is divisible by 5- if it ends in a 5 or a 0.
A number is divisible by 10 -if it ends in a zero.
Hints:
1. Underline the last digit of the number you are working with.
2. Determine if the number will be divisible by 2, 5, and 10 first because they are the
easy ones.
3. Next add up the digits in the number and write the sum in a circle next to your
number.
4. After you add the digits you can determine if the number will be divisible by 3, 6, and
9.