Download Rules of Divisibility

Math 6 Notes – Unit Four: Number Theory and Fractions
Syllabus Objective: (2.6) The student will solve number theory problems using primes,
composites, factors, multiples, and rules of divisibility.
Rules of Divisibility
It is beneficial for students to be familiar with rules of divisibility. When familiar with
these rules, it allows students to be focused more on the concept and skill of the lesson
rather than to be bogged with the arithmetic. Students already know some of these rules.
Chances are students already know if a number is divisible by 2, 5 or 10. For instance, if
asked to determine if a number is divisible by two, students can tell you that it has to be
even. Divisibility rules for 10 and 5 are also familiar to students.
RULES OF DIVISIBILITY: A number is divisible
By 2, if it ends in 0, 2, 4, 6 or 8
By 5, if it ends in 0 or 5
By 10, if it ends in 0
By 3, if sum of digits is a multiple of 3
By 9, if sum of digits is a multiple of 9
By 6, if the number is divisible by 2 and by 3
By 4, if the last 2 digits of the number is divisible by 4
By 8, if the last 3 digits of the number is divisible by 8
Example:
Is 111 divisible by 3?
One way of finding out is to divide 111 by 3; if there is no remainder, then 3 goes into
111 evenly. In other words, 3 is a factor of 111.
Rather than doing that, use the rule of divisibility for 3. Does the sum of the digits of 111
add up to a multiple of 3? Yes it does, 1 1 1  3 , so 111 divisible by 3.
Example:
Is 471 divisible by 3?
Use the rule of divisibility: Add the digits. 4 + 7 + 1 = 12. 12 is a multiple of 3,
therefore 471 is divisible by 3.
Example:
Is 12,316 divisible by 4?
Using the rule for 4, look at the last 2 digits, 16. Since 16 is divisible by 4,
12,316 is also divisible by 4.
Learning the Rules of Divisibility today will make life for students a lot easier in the
future. Not to mention it will save time and allow students to do problems very quickly
when other students are experiencing difficulty.
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 1 of 16
Prime Factorization
Prime Number: A number that is divisible by one and itself.
Examples:
2, 3, 5, 7 and 11 are all only divisible by 1 and itself.
Composite number: A number that is divisible by more than two numbers.
Examples:
4, 6, 8, 9, and 10 are all divisible by three or more numbers. For instance,
8 is divisible by 1, 2, 4 and 8.
The numbers 0 and 1 are neither prime nor composite numbers
Factor: Numbers that are multiplied to find a product.
Prime factorization: Rewriting a number as a product of prime numbers.
Example:
Write the prime factorization of 12.
12  4  3 . That’s a product, but 4 is not prime. So rewrite 4 as 2  2 .
12  4  3
12  2  2  3
12  22  3
Factor Trees
A factor tree is a systematic way of prime factorizing larger numbers one step at a time.
A composite number has exactly one prime factorization, however, the factor tree that
leads to that prime factorization may look different.
Example:
Write the prime factorization of 350 using a factor tree.
350
350
35
5
or
10
7 5
2
70
5
5
14
2 7
Looking at both factor trees above, the prime factorization for 350 is 2  5  5  7 or
2  52  7 .
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 2 of 16
The standard convention for writing a number as prime factors is to write the factors from
smallest to largest. However, it is not wrong if you do not. The preferred way to write
350 as a product of primes is 2  52  7 . But it could have been written as 7  52  2 .
Greatest Common Factor
Common factor: A number that is a factor of two or more nonzero numbers.
Example:
Find common factors of 18 and 24.
Factors of 18:
1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
1, 2, 3, and 6 are
the common factors
of 18 and 24.
Greatest Common Factor (GCF): Factors shared by two or more numbers are called
common factors. The largest of the common factors is called the greatest common factor.
In the last example, the greatest common factor, GCF, of 18 and 24 is 6.
There are a number of ways of finding the GCF.
STRATEGY 1
To find the GCF, list all the factors of each number. Look at the common factors and the
largest one is the GCF.
Example: Find the GCF of 24 and 36.
Factors of 24
Factors of 36
1, 2, 3, 4, 6, 8, 12, 24
1, 2, 3, 4, 6, 9, 12, 18, 36
The GCF is the greatest factor that is in both lists; 12.
STRATEGY 2
To find the GCF, write the prime factorization of each number and identify which factors
are in each number.
Example: Find the GCF of 24 and 36.
36 = 2  2  3  3
24 = 2  2  2  3
It might be easier for
students to see if the
common prime factors are
circled.
Each number has two 2’s and one 3, therefore the GCF = 2  2  3  12
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 3 of 16
Decimals and Fractions
Syllabus Objective: (3.4) The student will identify equivalent expressions between
fractions and decimals.
Batting averages are great examples of how decimals and fractions are related. For
instance, if Joe goes up to bat 10 times and gets 3 hits, his batting average is 3 10 or .300.
If Jane goes up to bat 8 times and gets 3 hits, her batting average is 3 8 or .375. This
example shows students that decimals or fractions can be used to represent the same
number.
It is also helpful to have students learn and memorize the following common
fraction/decimal conversions. Have students create flashcards for the following and refer
to them throughout the year as LTMR activities. If students are able to recognize and
convert these common fractions and decimals, it will increase their confidence and math
skills.
Common fraction
1
 .5
2
1
 .3333
3
1
 .25
4
1
 .2
5
1
=.1111
9
2
 .6666
3
3
 .75
4
2
=.4
5
2
=.2222
9
1
 .125
8
3
=.6
5
3
=.3333
9
decimal conversions:
4
=.8
5
4
=.4444
9
5
=.5555
9
6
=.6666
9
7
=.7777
9
8
=.8888
9
Converting Decimals To Fractions
Algorithm for:
Converting decimals to fractions
1. Determine the denominator by counting the
number of digits to the right of the decimal point.
That place value will be the denominator.
2. The numerator is the number to the right of the
decimal point.
3. Reduce.
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 4 of 16
Examples:
1) Convert .52 to a fraction. (52 hundredths)
Since 52 hundredths has two digits to the right of the decimal , the
denominator will be 100. The numerator is 52.
52
100
13

25
.52 
2) Convert .613 to a fraction. (613 thousandths)
.613 
613
1000
This fraction does not reduce.
3) Convert 8.32 to a fraction. (8 and 32 hundredths)
Since there are two digits to the right of the decimal, the denominator will be
100. The numerator is the number to the right of the decimal point, 32.
32
100
8
8
25
8.32  8
Converting Fractions to Decimals
One way to convert fractions to decimals is by making equivalent fractions.
Example:
Convert
1
to a decimal.
2
Since a decimal is a fraction whose denominator is a power of 10, look for a power of 10
that 2 will divide into evenly.
1 5

2 10
Since the denominator is 10, the decimal will end in the tenths place which means there
will be only one digit to the right of the decimal point. The answer is .5.
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 5 of 16
Example:
Convert
3
to a decimal.
4
Again, since a decimal is a fraction whose denominator is a power of 10, look for powers
of 10 that will divide into evenly. 4 does not go into 10, but will go into 100. By making
equivalent fractions,
3 75

4 100
There are denominators that will never divide into any power of 10 evenly. If that
happens, look for an alternative way of converting fractions to decimals. Could you
recognize numbers that are not factors of powers of ten? Using your Rules of
Divisibility, factors of powers of ten can only have prime factors of 2 or 5. That would
mean 12, whose prime factors are 2 and 3 would not be a factor of a power of ten. That
means that 12 will never divide into a power of 10. The result of that is a fraction such as
5 will not terminate – it will be a repeating decimal.
12
Because not all fractions can be written with a power of 10 as the denominator, it is
necessary to look at another way to convert a fraction to a decimal. That is, to divide the
numerator by the denominator.
Example:
Convert
3
to a decimal.
8
Since 8 is not divisible into a power of 10 number (only prime factor is 2), divide.
.375
8 3.000
The answer is .375.
Equivalent Fractions
Syllabus Objective: (3.3) The student will use models to translate among fractions and
decimals.
People have a way of describing the same thing in different ways. For instance, a mother
might say her baby is twelve months old, but the father might tell somebody his baby is a
year old. The age is the same but described in two different ways. Well, we do the same
thing in math; or in our case, fractions. Let’s look at the two cakes.
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 6 of 16
One person might notice that 2 out of 4 pieces seem to describe the same thing as 1 out of
2 in the picture above. In other words, 1 2  2 4 .
When two fractions describe the same thing, we say they are equivalent fractions.
Equivalent fractions are fractions that have the same value.
It would be nice to be able to determine if fractions were equivalent without drawing
pictures every time. Take a look at the following examples of equivalent fractions. Is
there a pattern?
Examples:
1 3

2 6
3 6

4 8
2 10

3 15
3 30

5 50
Is there a relationship between the numerators and denominators in the first fraction
compared to the numerators and denominators in the second fraction?
Notice both the numerator and denominator is being multiplied by the same number to
get the 2nd fraction.
Example:
5 20

if you multiply both the numerator and denominator by 4.
6 24
Since there is a pattern, there is an algorithm or procedure for generating equivalent
fractions:
To generate equivalent fractions, multiply BOTH numerator and denominator by the
SAME number
In the above example, when multiplying both the numerator and denominator by the
same number, we are multiplying by 4 4 or 1. When a fraction is multiplied by 1, it does
not change the value of the original fraction.
Example:
Express
5 ?

6 60
5
as sixtieths.
6
What do you multiply 6 by to get 60 in the denominator?
By 10, so multiply the numerator by 10.
5 50

.
6 60
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 7 of 16
Example:
2
is equal to how many thirty-fifths?
7
2 ?
What do you multiply 7 by to get 35 in the denominator?

7 35
By 5, so multiply the numerator by 5.
2 10

7 35
Mixed Numbers and Improper Fractions
Types of Fractions
1.
A proper fraction is a fraction less than one. The numerator is less than the
17
denominator. Example:
20
2.
An improper fraction is a fraction greater than one. The numerator is greater than
31
the denominator. Example:
20
Converting Mixed Numbers to Improper Fractions
A mixed number is a whole number greater than 1 and a fraction. A mixed number is
used when there is more than one whole unit. For example, 1 1 4 is called a mixed
number.
From the picture above, let’s say Johnny ate the entire first cake and one piece from the
second cake. This could be described as eating 1 1 4 cakes. His mom might come home
and notice Johnny ate 5 pieces of cake. Notice, eating 1 1 4 cakes describes the same thing
as eating 5 pieces of cake.
Since we are working with fractions, 5 pieces of cake can be described as a fraction. The
numerator tells how many pieces Johnny ate and the denominator tells how many equal
pieces make one whole cake. In this case the fraction then is 5 4 .
It seems that 1 1 4 describes the same thing as 5 4 . Therefore, it can be stated that they are
equivalent; 1 1 4  5 4
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 8 of 16
Example:
The above diagram represents two cakes and the shaded regions indicate the
pieces that have been eaten. It looks like an entire cake has been eaten plus 3 8 of
the second cake. This is represented by the mixed number 1 3 8 cakes.
Another way to describe the eaten pieces is to state that 11 pieces of cake have
been eaten from cakes that are cut into eight equal pieces. This can be written as
11 . Therefore it can be said that 1 3 is equivalent to 11 .
8
8
8
Ask students if there is a way to convert mixed numbers to improper fractions
without having to draw cakes for every problem.
One whole cake can be written as 1 or
8
8
and the portion of the other cake is
3
8
.
3
1 
8
3
1 
8
8 3
 
8 8
11
8
Ask students if they can see a pattern.
1
1 5
3 11
 and 1 
4 4
8 8
Algorithm for:
Converting a mixed number to an improper fraction
1. Multiply the whole number by the
denominator
2. Add the numerator to that product
3. Place that result over the original
denominator
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 9 of 16
Example:
Convert 2 43 to an Improper fraction.
1. Multiply 2  5 , which is equal to 10.
2. Add 3 (numerator) + 10, which is equal to 13.
13
3. Place 13 over the original denominator:
.
4
Converting Improper Fractions to Mixed Numbers
From previous examples:
5
1
1
4
4
11
3
1
8
8
13
3
2
5
5
Students are now going to find an algorithm for converting improper fractions to mixed
numbers.
Example:
5 4 1
1
1
   1  1
4 4 4
4
4
This means that the above improper fraction can be re-written as the sum one
whole cake plus a fraction of a cake.
Example:
13 5 5 3
3
3
    2  2
5 5 5 5
5
5
This means that the above improper fraction can be re-written as the sum of two
whole cakes plus a fraction of a cake.
The trick to convert an improper fraction to a mixed number seems to be is to
determine how many whole cakes were eaten, then write the fractional part of the
cake left.
Example:
To convert 7 3 to a mixed number, how many whole cakes were eaten? Well 7 3
could be written as 3 3  3 3  1 3 . Two whole cakes were eaten plus a 1 3 of
another cake. Therefore 7 3 converted to a mixed number is 2 1 3 .
Can this be done without breaking apart the fraction?
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 10 of 16
To determine how many whole cakes there are in 7 3 , divided 7 by 3. The quotient is 2,
which means there are 2 whole cakes. The remainder is 1, which means there is one
piece of the last cake, or 1 3 of a cake left. This means 7 3  2 1 3 .
Algorithm for:
Converting an improper fraction to a mixed number
1. Divide the numerator by the denominator to
determine the whole number
2. Write the remainder over the original
denominator
Example:
Convert 22 5 to a Mixed Number.
1. 5 goes into 22 four times.
2. The remainder is 2.
22
2
4
5
5
Comparing and Ordering Fractions
Syllabus Objective: (1.1) The student will order fractions and decimals.
To compare fractions with different denominators, find a common denominator, make
equivalent fractions, and then compare the numerators.
Common Denominators
Let’s say there are two cakes, one chocolate, the other vanilla. The chocolate is cut into
thirds and the vanilla into fourths as shown below. Ashton eats one piece of chocolate
cake and one piece of vanilla cake, as shown by the gray pieces.
Since two pieces of cake have been eaten, is it correct to say that Ashton ate
cake?
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
2
7
of a
Page 11 of 16
Let’s re-define a fraction. The numerator tells how many equal pieces have been eaten
and the denominator states how many equal pieces make one whole cake. Since the
pieces are not equal, 2 7 of a cake does not fit the above definition of a fraction, and
clearly, 7 pieces do not make one whole cake: Therefore, trying to add 1 4 to 1 3 and
coming up with 2 7 does not fit the definition of a fraction.
The key is to cut the cakes
into equal pieces.
The dark lines indicate
additional cuts to each cake.
By making additional cuts on
each cake, both cakes are now
made up of 12 equal pieces.
That’s good news from a
sharing standpoint – everyone
gets the same size piece.
Mathematically, the concept of a common denominator has just been introduced.
Here is the way the additional cuts were made: Cut the second cake the same way the
first was cut and the first cake the same way the second was cut as shown in the picture.
Now, that’s a piece of cake!
Clearly, it is not convenient to make additional cuts in cakes every time there are cakes
with unlike pieces. It is necessary to find a way that will allow students to determine how
to make sure all the pieces are the same size. That process is called “finding a common
denominator.”
Three Methods of Finding a Common Denominator
A common denominator
is a denominator that all
other denominators will
divide into evenly.
1. Multiply the denominators
2. Write multiples of each denominator, find a common multiple
3. Use the Reducing Method, especially for larger denominators
Cake-wise, it’s the number of pieces that cakes can be cut so everyone has the same size
piece.
Method 1: Find the common denominator between 1 3 and 1 4 .
Multiply the denominators. The common denominator would be 3  4 or 12 .
Method 2: Find the common denominator between 2 3 and 1 4 .
Write out multiples of 3 and multiples of 4. Find a multiple that is in
common to both denominators.
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 12 of 16
Multiples of 3: 3, 6, 9, 12, 15, 18, …
Multiples of 4: 4, 8, 12,…
Since 12 is a multiple of each denominator, 12 would be a common
denominator.
Method 3: This method is used when the denominators are large and methods 1 and 2
are not very convenient.
Find the common denominator between
5
18
and
7
24
.
Use the two denominators, 18 and 24, to write the fraction then reduce the
fraction:
18 3

24 8
Now cross multiply, either 24  3 or 18  4. It does not matter which way
the cross multiplication occurs, the product is 72; therefore the common
denominator is 72.
Notice it does not matter whether the ratio of the denominators is 18 24 or 24 18 .
After reducing and cross multiplying, the common denominator is still 72.
Example: Find the common denominator for
5
9
and
.
24
42
While multiplying will give you a common denominator, that product will be
very large. Use method 3, place the denominators over each other and
reduce.
24 4

42 7
42  4 or 24  7 gives a product of 168. The common denominator is 168.
Example: Use > or < to compare the fractions.
.
3
2
5
3
Find a common denominator and make equivalent fractions, then
compare the numerators:
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 13 of 16
3
5
2
3
Since 9 < 10, we have
Example:
9
15
10
15
3
2
< .
5
3
Order the following fractions from least to greatest.
3 7
2
, ,and
4 10
3
To order the three fractions, find a common denominator for all three
fractions and make equivalent fractions using the common denominator.
Then, compare the numerators to re write the fractions from least to greatest.
The common denominator is 60. Methods 1 and 3 are not practical to find
the common denominator of three fractions. Use method 2 to find the
common denominator. Have students notice that one of the denominators is
10. Which multiple of 10 is also a multiple of 4 and 3? 60 is the first
multiple that all three denominators have in common.
Using 60 as the common denominator, make equivalent fractions:
3 is equivalent to 45
7 is equivalent to 42
2 is equivalent to
60
10
60
4
3
40
60
Compare the numerators and rewrite the fractions from least to greatest:
40 42 45
2 7 3
,
,
; which are equivalent to the answer of , , .
60 60 60
3 10 4
Adding/Subtracting Fraction With Like Denominators
Syllabus Objective: (3.1) The student will calculate with rational numbers expressed as
fractions and mixed numbers.
In order to add or subtract fractions, we have to have equal pieces. Suppose there is a
cake cut into 8 equal pieces. Three pieces are eaten by Ashton and four pieces are eaten
by Catherine. This means 7 pieces of cake, or 7 8 of one cake has been eaten.
Mathematically, we would write 3 8  4 8  7 8 .
Notice, only the numerators are added because the numerator represents how many equal
pieces are eaten. Why are the denominators not added? Remember that fractions are
defined where the denominator tells how many equal pieces make one whole cake. If the
denominators were to be added that would mean there are 16 equal pieces of cake, and
that is not the case for this example.
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 14 of 16
Algorithm for:
Adding/Subtracting fractions with like denominators
1. Add the numerators, keep the same
denominator.
2. Write your answer in simplest form.
Example:
Subtract
3
7
3
1 .
10 10
Subtract the whole numbers: 3  1  2
7
3
4


Subtract the fractions:
10 10 10
4 2

Simplify:
10 5
The answer is: 2
2
5
Borrowing when subtracting mixed numbers:
The concept of borrowing when subtracting with fractions has been typically a difficult
area for kids to master. For example, when subtracting 12 1 6  4 5 6 , students usually
answer 8 4 6 if they subtract this problem incorrectly. In order to ease the borrowing
concept for fraction, it would be a good idea to go back and review borrowing concepts
that kids are familiar with.
Example: Take away 3 hours 47 minutes from 5 hours 16 minutes.
5 hrs 16 min
 3 hrs 47 min
?????????
Subtracting the hours is not a problem but students will see that 47 minutes cannot
be subtracted from 16 minutes. In this case, students will see that 1 hour must be
borrowed from 5 hrs and added to 16 minutes:
4hrs 5 hrs 16 min16min 1hr  16min 60min  76min
 3 hrs 47 min
?????????
Now the subtraction problem can be rewritten as:
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 15 of 16
4 hrs 76 min
 3 hrs 47 min
???????????
4 hrs 76 min
 3 hrs 47 min
1 hr 29 min
If students can understand the borrowing concept from the previous example, the same
concept can be linked to borrowing with mixed numbers. Lets go back to the first
example: 12 1 6  4 5 6 . It may be easier to link the borrowing concept if the problem is
1 1
1 6 7
11 12
1   
7
6 6 6
6 6
11
rewritten vertically:
6
5
 4
5
6
 4
???????
6
7
Example: Evaluate a 
2
1
7
6
3
5
1
for a  . Write the answer in simplest form.
18
18
5 1
6 1
 

18 18 18 3
Example: Solve. x 
5 4

9 9
x
9
1
9
Example: Catherine had 6 cups of chocolate chips. She used 2 1 3 cups to make cookies
and 1 2 3 cups to make pancakes. How many cups of chocolat3e chips does she have now?
The following are links to websites that offer instructions and games to reinforce fraction
skills:
http://www.nga.gov/education/classroom/counting_on_art/act_fractions.shtm
http://www.bbc.co.uk/skillswise/numbers/fractiondecimalpercentage/
Holt, Chapter 4, Sections 1-8
Math 6, Unit 4: Number Theory and Fractions
Revised 2012 - CCSS
Page 16 of 16