Download Divisibility Rules

Math6 1st Qtr Overview
Divisibility Rules
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: A number
is divisible by 12; if it is 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 of a given number. When the prime numbers are multiplied together equals the prime
number that was originally factored.
Begin by writing two factors that equal the # to prime, then do the same for each number till all #’s are prime.
EX:
96
Think of any two factors of 48—doesn’t matter what you start with the answer will be the same
2 x 48
2 is a prime number and does not have to factors other than 1 and itself (can’t put 1 in a tree)
2 x 24 Think of any 2 factors of 48 ---2 is prime
2 x 12
Think of any 2 factors of 12 ---2 is prime
3x4
3 is prime—4 has two factors
2 x 2 both 2’s are prime
When you multiple all the prime numbers to ether in any order you get a factor of 96!! Cool huh!
Prime factorization: 2 x 2x 2 x 2 x 2 x 3 x 4 = 96
Write the answer in exponential form: 2^5 x 3 x 4 = 96
LCM
Find the smallest multiple that other numbers can divide into evenly; It is the smallest MULTIPLE for both
numbers
GCF
Find the largest factor that will divide evenly into other numbers: It is the greatest FACTOR for both numbers
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; as long as it is the SAME
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 whole numbers—0, 1, 2, 3, 4 stays the same as it did before; 5, 6, 7, 8, 9 goes up
one every time!
 The rule number is the # to the immediate right of the number being rounded
 The rule number will become a 0 with all other number to the right of the number that was rounded
after the rounding decision has been made
 The number 56.4367 is a decimal even though it has a whole number (56); It would round to 56
o It is confusing to understand that you round only the whole number when rounding a decimal
 If you are told a specific place value such as the tenths, hundredths, thousandths, etc.. you round
only that digit in the decimal
 Remember all the number to the left of a rounded number do not change.
EX: .435 round to nearest hundredth; find hundredth place (3)…look to the right fort he rule number which is 5. 5
says to go up one; .44
78.289 round to nearest whole number or front-end estimation; find the whole number (78)…look to the right for
the rule number which is 2. 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 necessary with subtracting when regrouping is necessary.
Comparing Decimals
Compare the front-end whole number FIRST. The larger number wins! When comparing past the decimal, the
larger number wins there also. Begin comparing the numbers closest to the decimal then work outwards. 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 than only 499 of 1,000. Although they are very close .5 > .499
Don’t forget that when determining the < & > symbols that they are the tips of arrows pointing along a number line.
< points in the direction of a number line decreasing (less) and > is pointing in the direction a number line increases
…………………………………..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
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)
For EVERY Problem!
*To work a fraction problem the denominators have to be the same number!
*Always review answer. . .Ask yourself. . .
*Is it improper? If it is change it to a mixed number
Add don’t forget to add the whole # from the improper fraction you fixed to the one you already had
*Does it reduce?
Is there a factor that will divide evenly into the N & the D?
Remember: It will be the N or lesser!
Order of Operations
PEMDAS (Please excuse my dear Aunt Sally)
Parenthesis Work any portion of the problem in ( ) first;
A. FYI If there are more than one set of parenthesis or
parenthesis within brackets, work from the inside parenthesis outwards
B. Problems within the parenthesis have to be worked by the order of operations PEMDAS
C. A parenthesis is NOT complete until only one number comes out. Do not work any other problem
until only one number comes out of the parenthesis
Exponents
Multiply
Divide
Add
Subtract
Simplify exponential problems; The exponent tell you how many times to multiply the larger number
by itself. 6^3 (exponent comes after the ^) 6 x 6 x 6 = 216 (6 x 6 = 36 ; 36 x 6 = 216)
Multiply & Divide from L to R; NOT all multiplication problems then division problems
Scan problem from L to R and work the first multiplication or division sign you come to
Rewrite the problem then scan from L to R again and work the next multiplication or division sign
you come to; repeat
Add & Subtract from left to right; NOT all addition problems then subtraction problems
Scan problem from L to R and work the first addition or subtraction sign you come to
Rewrite the problem then scan from L to R again and work the next addition or subtraction sign you
come to; repeat