Download Divisibility Rules

Math6 ADV 1st Qtr Overview
Divisibility Rules
Focus on 2,3,4,5,6,10
2 (end in even number)
3 (sum of digits div. by 3)
4 (last two digits)
5 (ends in 0 or 5)
6 (div. by 2 & 3)
7 (double last digit; subtract sum from rest of #, if difference is div by 7)
8 (last 3 digit div. by 8) 9 (sum of digits div. by 9)
10 (ends in 0)
No rule # (If divisible by two unrelated factors of that number it is div by that number EX: 12; div by 3 & 4)
Prime & Composite Numbers
Prime number—number only divisible by 1 and itself
Composite number—number divisible by 1, itself, and at least one other number
Prime Factorization
Finding the prime numbers that when multiplied together equals the prime number factored
Begin by writing two factors that equal the # to prime, then do the same for each number till all #’s are prime
Write answer in exponential form EX; 2 2 2 3 3 4 5 would be written 2^3 X 3^2 X 4 X 5
LCM
Find the smallest multiple that other numbers can divide into evenly; It is the smallest MULTIPLE for both
GCF
Find the largest factor that will divide evenly into other numbers: It is the greatest FACTOR for both
Improper to Mixed
When the denominator (D) is less than the numerator (N),
Divide the D into the N to get a whole number; place the remainder as the N over the original D
Mixed to Improper
(D X whole number) + N—place over the original D
Equivalent Fractions
To get a larger equivalent fraction multiply the N & D by the same number (any number)
To get a lesser equivalent fraction divide the N & D by the same number (It must be a factor of both #’s)
_____________________All the above is needed to solve fraction problems____________________
Converting Fractions, Decimals, & Percents (Percents are not necessary 1st quarter.
Focus on fraction to decimal and decimal to fraction)
F to D
F to P
D to F
D to P
P to D
P to F
Divide the denominator into the numerator using the decimal rule
(FDP) Change the fraction to a decimal (as above); then move the decimal 2X to the right, add a %
Decimal over a fraction bar; for D, write 1 then a 0 for each digit past the decimal; remove decimal; reduce
Move the decimal sign 2x to the right; add a %
Move the decimal sign 2x to the left; remove %
(PDF) Change the percent to a decimal (as above); then D to F (as above)
Estimating Decimals
Use the same rule for rounding—0, 1, 2, 3, 4 stays the same as it did before; 5, 6, 7, 8, 9 goes up one every time!
EX: .435 round to nearest hundredth; find hundredth place (3)…look to the right, 5 says to go up one; .44
78.289 round to nearest whole number or front-end estimation; find whole number (78)…look to the right, 2 says
stay the same; 78
Add & Subtract Decimals
MINI LESSON: All whole numbers have an invisible . (decimal) at the end. We have always lined up decimals to
add & subtract to be sure our place values matched! EX: 324 + 21 ; 324. + 21. We lined the decimals up and the
ones place and tens placed lined up. Then we worked the problem!
Adding and subtracting with decimals is solved the same as whole numbers. Just line up the decimal (or invisible
decimal) before you work the problem. To help keep the numbers organized, once you line up the decimals, add
zeros in where there are missing digits. it is extremely helpful with subtracting when regrouping is necessary.
Multiplying Decimals
There is only 2 extra steps when multiplying decimals verses whole numbers.
Multiply as usual.
-Then count the number of digits to the right of the decimal in the entire problem! Remember how many!
-Begin at the end of the number, you got when you multiplied and move the invisible decimal to the left the same
number of times there are digits to the right of the decimal in the problem. Your done!
Dividing Decimals
If there is a decimal in your divisor then you CANNOT divide. There are no rules to divide decimals into whole
numbers or decimals. You have to move the decimal in the divisor to the end of the number to make it a whole
number. Then move the decimal in the dividend the same number of times. If you don’t see a decimal in the
dividend, remember it is invisible and at the end. Fill in with zeros if there are no numbers when you move the
decimal. NOW, divide as usual. The decimal with be in the quotient at the same place it is in the dividend!
Comparing Decimals
Compare the front-end first. The larger number wins! When comparing past the decimal, the larger number wins
there also. The confusing part can be that .5 is more than .499 ; What you have to remember is that the further a
number is written to the right of a decimal the SMALLER it becomes. 5/10 is larger than 499/1,000! I would
rather have 5 of 10 that only 499 of 1,000. Although they are very close .5 > .499
…………………………………………….FRACTIONS!................................................
Compare & order fractions
Use the LCM to get the same D, then compare the N. THAT’S ALL!!
Estimating Fractions (benchmark: 0, ½, 1)
Understand what a half looks like…If the numerator is half the amount of the dominator the fraction is equal to ½
Ex: 12/24 = 1/2
(N could only be 12); If it were 2 12/24, the fraction becomes a 1/2; 2 + 1/2 = 2 1/2
If the numerator is less than half of the denominator round to 0
Ex: 11/24 = 0
(N could be 11 through 1); If it were 2 11/24, the fraction becomes 0; 2 + 0 = 2
If the numerator is more than half of the denominator round to 1
Ex: 13/24 = 1 (N could be 13 through 23); If it were 3 13/24, the fraction becomes 1; 3 + 1 = 4
Multiplying Fractions
You do not use an LCD when multiplying fractions
Multiply a whole number to a fraction
Multiply a fraction to a mixed number
Multiply whole number to a mixed number
Multiply mixed numbers to mixed numbers
 Before multiplying set up the problem correctly
 Place any whole numbers over a 1
EX: 67 --67/1
 Change a mixed number to improper
 Follow the steps to ‘multiply fractions’ below
9-- 9/1
Multiply Fractions
 Ex: 4/25 x 5/12
 First, cross simplify by finding a factor for the N on one side and the D on the other
o the GCF for 4 and 12 is 4 4 divides into 4 1 time; 4 divides into 12 3 times
o Rewrite as 1/25 x 5/3
o Check to see if there is a GCF for 5 and 25; the GCF is 5; 5 divides into 5 1 time
and 5 divides into 25 5 times
o Rewrite as 1/5 x 1/3
 There is not always a GCF for both side and sometimes there is not a GCF for either
o If you do not cross simplify or use the GCF the answer will still come out correct
as long as you do not make a numeric error, but you will have to simplify more at
the end
 Second, multiply the N to N and the D to D
 Third, be sure your answer is in the proper form
Dividing Fractions
Divide a whole number to a fraction
Divide a fraction to a mixed number
Divide whole number to a mixed number
Divide mixed numbers to mixed numbers
You do not use an LCD when multiplying fractions
You CANNOT divide fractions! The problem must be changed to a multiplication problem to
work it
 Before dividing set up the problem correctly
 First, place any whole number over a 1
EX: 67 --67/1
9 -- 9/1
 Second, change a mixed number to improper
 Third, stay-change-flip
o STAY - This means to leave the first fraction as is
o CHANGE – the operation from division to multiplication
o FLIP – Flip the second fraction—place the D in the N position and the N in
the D position (It is now a multiplication problem)
 Fourth, cross simplify if possible (Not always possible)
 Fifth, multiply N to N, D to D
 Sixth, Change from improper to mixed
 Seventh, Reduce if necessary
(Not always necessary)
LIKE DENOMINATORS:
With like denominators, there is NO NEED to get a LCD. The Denominators are already the same!!
Add Fractions w/ like denominators
Leave the denominator the same and add the N
Sub Fractions w/ like denominators
Leave the denominator the same and subtract the N
Add Mixed Number w/ like denominators
Leave the denominators the same and add the N, then add the whole #’s
Subtract Mixed Number w/ like denominators
Leave the denominators the same and subtract the N, then add the whole #’s
UNLIKE DENOMINATORS:
Add Fractions w/ unlike denominators
Sub Fractions w/ unlike denominators
1-Find LCD w/ LCM
Check to see if LCM is the largest D; It is if it is divisible by the least D
If not, write the multiple of the largest D until there is a multiple divisible by the least D—that’s the LCD
2-Once the LCD is found get the new N by
Dividing the old D into the New LCD, multiple the quotient by the old N to get the New N
3-Solve by adding or subtracting the N
4-Always check to see if the problem is in the proper form
Will it reduce? Is there a factor that will divide into both the N or D
Is it improper? Divide the D into the N—quotient is the whole number, remainder is the N, divisor is the D
Add Mixed Numbers w/ unlike denominators
Sub Mixed Numbers w/ unlike denominators
*Same as above with the exception
Add or subtract mixed numbers
If the answer is improper, the whole number from the new mixed number is added to the whole number
already there
Sub Mixed Numbers w/ unlike denominators w/ regrouping
*Same as above with the exception
 The N on the top fraction will not be large enough to subtract the lower N
 Regroup: borrow a 1 from the whole number of the lesser fraction
 Change that 1 into fraction form of a one
o The form the 1 should take is the same number as the D of the lesser fraction
 Add the fraction 1 that was borrowed to the lesser fraction
 Now SUBTRACT!!
Sub Mixed number and whole numbers w/ regrouping



When there is a whole number in a fraction problem that a mixed number is to be taken from
borrow a 1 from the whole number
change the 1 to a fraction that is formed from the D in the mixed number; then subtract as usual
Once a fraction problem is worked the answer has to be put in the proper
form:
Examples of answers to put in the correct form
A.
Reduce
36/48
Ask yourself if the least number (36) can divide evenly into the D. If yes the GCF is 36.
If no...mentally...try all numbers less than 36 to 2. The GCF is the largest number that will divide evenly into both
the N and D evenly; The GCF is 12
¾ is the reduced answer
B.
Reduce w/ whole number 4 2/8 Reduce only the fraction and rewrite the answer with the same whole
number (4 ¼)
C.
Improper 9/7 Improper fraction is changed to a mixed number; divide D into N; Place the number of whole
times the D divides into the N as the whole number, the remainder as the N and the D is the same D: (1 2/7)
D.
Improper w/ whole number 6 5/3 Change only the fraction to a mixed number ( 1 2/5); then add the
whole number that came from the fraction to the whole number you had (final answer 7 5/3)
E.
Improper that will reduce w/ whole number D & B above
12 9/6 change improper fraction only to 1 3/6; 3/6 will reduce, reduce only the fraction to 1/3; 12 is
still whole number in answer and so is the 1 (12 + 1 =13) that is the whole number and the fraction is 1/3
(final answer 13 1/3)
Ratios
A ratio is a way to represent a relationship between two things.
A ratio can be written in 3 forms:
5/8
(like a fraction)
5 to 8 (with the word ‘to’)
If rewritten as a ratio in a more friendly way to solve a problem the first number is
the top number
5:8
(with a colon)
Unit Rate
A ratio that compares two quantities measured in different units
A unit rate finds the cost of one unit
EX: There are 92 desks in 4 classrooms. The unit rate would be how many desks are in 1 classroom. 92/4
 Divide the 4 into the 92 to find 23 classrooms per room
Proportions
Setting two ratios equal to each other
If two ratios are equal then their cross products will be equal
 cross products: the numbers diagonal from each other multiplied together will have the same
product
EX: There are 9 players per team. Find the number of players on 7 teams.
Notice that you only see two numbers. Per team means 1 team. 1 is the other number.
Write the number related to each other in the order you see them.
9 players/1 team = ? players/ 7 teams
Solve by multiplying the number that are diagonal from each other and divide by the other
 9 by the 7 and divide by the 1
Ex: It cost $29.85 for 3 presents. How much will it cost for 11 presents?
29.85 cost /3 presents = ? cost/ 11 presents
 29.85 x 11 $328.35
divided by 3 = $109.45 is the answer
Finding a Percent of a number
EX: 36 % of 67
 Set problem up as a proportion 36/100 X 67/1
 36 is placed over a 100 because 36% is some part of 100 %
 You cannot work with a % sign in a problem and 36% was converted to a decimal by moving the
decimal two time to the left (away from the % sign). .36 converted to fraction is 36/100
 36/100 of 67 is the new problem
 To multiple a whole number by a fraction the whole number has to be converted to a fraction
 The 67 is placed over the ‘invisible 1’
 36/100 of 67/1. You cannot cross-simplify this problem
 Multiply straight across. 36 X 67 = 2432 100 X 1 = 100; 2432/100 improper to mixed is 24.32

24.32 is 36% of 67.