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36 Cards in this Set

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Divisibilty Rules:

An integer is divisible by:
2 if the integer is even

3 if the sum of the integers digits is divisible by 3

4 if the integer is divisible by 2 twice

5 if the integer ends in 0 or 5

6 if the integer is divisible by both 2 and 3 (i.e. it is even and the sum of the digits is divisble by three)

8 if the number is divisble by 2 three times

9 if the sum of the digits are divisble by nine

10 if the integer ends in 0
zero is
a multiple of every integer
a factor is a
`POSITIVE integer that divides evenly into an integer. so 1, 2, 4, 8 are all factors of 8
a multiple of an integer is
formed by multiplying the integer by any whole number... including zero.
Sum and Difference Rule
If 2 numbers have a common divisor/factor there SUM and DIFFERENCE retain the common divisor. I.E. 8 is a factor of 40 and 64. 8 is also a factor of 64-40=24 and 64+40=104
y=xq+r
104=8*13
24=8*3

y=dividend
x=divisor
q=quotient
r=remainder
primes
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199
factor formula:

how many factors does 18 have
step one break into primes:

2 * 3^2

add one to each power and multiply through

2*3=6 factors

long way:
1 * 18
2 * 9
3 * 6
factor foundation rule:
if an integer is divisible by some number n, it is also divisable by all factors of n.

i.e. if 72 is divisble by 12, it is also divisble by 1,2,3,4,6)
Is 27 a factor of 72?
Procedure:

break 27 into primes:
3^3

break 72 into primes

2^3 3^2

we can not make 27 from the prime factors of 32 therefore 27 is not a factor of 72
Given that the integer n is divisble by 3, 7, and 11 what other numbers must be divisors of n
since we know that n has AT LEAST 3,7, and 11 as primes we can say that it is also divisble by 3*7=21 3*11=33 11*7=77
and 3*7*11=231
Lowest Common Multiple
the smallest number that is a multiple of two integers

procedure:

us a prime box and find the product of all the prime numbers in each box, using the higher power of any primes that might appear in both boxes
Greatest Common Factor:
the largest number by which two integers can be divided

procedure:

us a prime box, and find the product of the primes that are common to both numbers to the lower power
What is the GCF/LCM of 24 and 30
24=
2^3, 3

30=
2,3,5

GCF = 2*3=6
LCM = 2^3*3*5=120
terminating decimals
if the denominator has only factors of 2 or 5 you are garunteed a terminating decimal

note 2^0 or 5^0 still counts

ie a denominator of 5, 2, 25
if 80 is a factor of r is 15 a factor of r
PROCEDURE:

USE PRIME
BOX

80 = 2^4*5

15=3*5

SINCE 3 IS NOT IN THE PRIME BOX OF 80 WE CAN NOT MAKE 15 OUT OF THE FACTORS OF 80... SO WE DO NOT KNOW IF 15 IS A FACTOR OF R (IT COULD BE) REMEMBER THAT THE PRIME BOX IN
THIS SITUATION COULD REPRESENT ONLY A PARTIAL LIST OF R'S FACTORS.
given that 7 is a factor of n and 7 is a factor of p is 7 a factor of n+p
yes. use the sum and difference rule. if x is a factor of integers n and m, x will be a factor of n+m and n-m
ODDS AND EVENS
Odd +/- Odd = Even
Odd +/- Even = Odd
Even +/- Odd = odd.
Even +/- Even = Even

Odd X Odd = Odd
Odd X Even = Even
Even X Even = Even
-|x| = x for all x <=0


for x = -5

-|-5| = -5
also |x| = -x for all negative x

x=-5

abosolute value of x = 5, 5 = -(-5)
what is 66 divided by -33 times |-9|
be careful of wording or text as sometimes it is easy to think this problem means


66/(-33*|-9|) but it really means

(66/-33)*|-9|

the answer is -2*9 = -18
consecutive integers follow one another without skipping from an even starting point
consecutive Integers:

-1,0,1,2

consecutive even

0,2,4

consecutive primes

2,3,5,7...
how many integers are between 14 and 765 inclusive
procedure:

subtract and add one 765-14+1=752
how many factors of seven are there from 8 49 inclusive
arithmetic progression formula

an = a1 + (n-1)d

an= nth term >> end number
a1= first term >> first number divisable by factor
n= number of terms
d= common difference >> factor

49 = 14 +(n-1)7
49 = 14 +7n -7
n =6 six multiples
How many integers between 100 and 150, inclusive, cannot be evenly divided by 3 nor 5
there are (150-100)+1 = 51 numbers between 100 and 150 inclusive.

using Arithmetic Progression formula:

150 = 105 + (n-1)*15 (numbers that are devisable by 3*5) n= 4
150 = 102 + (n-1)*3 (numbers that are devisable by 3) n = 17
150 = 100 + (n-1)*5 (numbers that are devisable by 5) n = 11

since we are looking for numbers that cannot be evenly divided by 3 nor 5, then:

51-17-11+4 = 27
sum of consecutive integers (ie 2,3,4... only integers not consecutive even, odd, etc...)

what is the sum of all the integers from 20 too 100 inclusive
1: find the middle:

(100+20)=60

2:
find the number of terms:

100-20+1=81

multiply the average by the number of terms:

81*60=4860
what is the average term in a consecutive set?

what is te average of all the integers from 20 to 100 inclusive?
if you know only first and last term:
(100+20)/2 = 60

if you know the sum of the set and number of terms in the set:

sum of all integers/# of terms = average

4180/81 =60
The product of n set of consecutive integers will always have n as a factor.
1*2*3 = 6
2*3*4 = 24

both have 3 has factor

5*6*7*8= 1680 is divisible by four
Special Sums:

For any set of CONSECUTIVE INTEGERS with an odd number of terms, the sum of the terms is always a multiple of the # of terms
1+2+3=6
8+9+10=27

both multiple of 3

4+5+6+7+8=30
13+14+15+16+17=75

both multiples of 5.

THIS APPLIES ONLY TO A SET OF CONSECUTIVE INTEGERS WITH AN ODD NUMBER OF TERMS.

consider three consecutive integers:

n, n+1, n+2
=3n+3 is divisble by 3

4n+6 is not
if x^3-x=p and x is even is p divisble by 4?
x-1=odd
x=even
x+1=odd

factor the problem

x(x^2-1)
x(x+1)(x-1)
e * o * o
answer:

we know it is even but we can not say for certain it is divisable by 4.
list six factors of the product of five consecutive even integers
all we can say is that it has a prime box of 5 2's

so

2
2*2=4
2^3=8
2^4=16
2^5=32
beware of negative exponents as they can hide the the base
x^2=4

x= -2 or 2
x^2 = x^6 = x^8

x=?
x = 0,1,or -1
(3^6)^2=
(x^a)^b
3^12
x^(ab)
when can you combine exponents?
if two expressions have a base in common or an exponent in common you can combine them and ONLY for multiplication and division:

SAme Base:

x^a * x^b = x^a+b
3^2 * 3^3 = 3^5

x^a/x^b = x^a-b
3^3/3^4 =3^-1 = 1/3

Same exponent:
x^a*b^a=xb^a
3^3 * 5^3 = 3*3*3*5*5*5
15^3

x^a/b^a = (x/b)^a
9^3/3^3=
(9*9*9)/(3*3*3)=3*3*3 =3^3
When can you combine squareroots
during multiplication you can split larger roots into separate factors:

sqrt400 =sqrt(25*16) =5*4=20

During division you can split a larger quotient into dividend and remainder

sqrt(144/16) = sqrt144/sqrt16
=12/4=3

YOU CAN NOT SPLIT OR COMBINE ADDITION/SUBTRACTION
Imperfect squares do not yield an integer
you can estimate:

sqrt(52) is between sqrt49 and sqrt64 i.e. between 7 and 8

or:

sqrt52 = (sqrt4*13) = 2sqrt13