Integer Properties Flashcards
What are real numbers?
Any number that can be found on a number line
What are imaginary numbers?
Numbers that don’t exist
Whats an example of an imagery number?
A square root of any negative number
What number can be squared to obtain -4?
None
What is an integer?
Any number that doesn’t contain a decimal or fractional part
I.e. 1,2,3 and -1,-2,-3
Is 0 an integer?
Yes
Is 0 even or odd?
Even
What is n\0?
Undefined
Can you divide by 0 ?
No
Whats another name for divisors?
Factors
Can negative numbers be factors?
No
0 is a multiple of every integer
….
Can multiples be negative?
Yes
Whats are some multiples of 3?
-3,0, 3, 9
Is 0 a factor of any number?
No because anything divided by 0 is undefined
Every positive integer is both a factor and a multiple of itself
I.e. 5 is factor and multiple of itself
What is a trick to figuring out the number of factors an integer has has?
Approximate the square root of that integer and apply the divisibility rules for every integer equal to or less than the square root
I.e. to find factors of 80 think about the closes number is 80 whose perfect square is 9 so do all divisibility rules to 9
What is a prime number?
A positive integer that only has two factors
Is 1 a prime number? Why or why not?
No, its only factor is itself. Prime numbers have two factors.
How many prime numbers are there up until 40?
12
What are the prime numbers up until 40 ?
2, 3, 5, 7, 11, 17 , 19, 23, 29, 31 ,37,
What is the only even prime number?
2
How many prime numbers are there?
An Unlimited amount
What must be true for the product of two integers to be odd?
Both integers must be odd
What happens when you multiply a negative number by a positive number ?
You get a negative number
What happens when you multiply two negative or two positive numbers?
Your outcome is positive
A negative number cubed will result in …
A negative number
Any even number minus 1 will produce…?
An odd number
If r and s are prime numbers can (r/s) be an integer ?
No because prime numbers are only divisible by 1 and themself
How can you test numbers under 100 to see if they’re prime ?
Test the numbers w the divisibility rules of 2,3,5,7
Is 97 prime ?
Yes
How do you answer a problem that ask you to identify one or more numbers that compromise the product ?
Do prime factorization of the number
How do you answer the question how many unique prime numbers between 1 and 50 are factors of 16,900?
Do prime factorization
See how many different prime factors the number has under 50
The answer is 3
How do we determine the total number of factors of any integer ?
Determine the prime factorization of the integer
Add 1 to each exponent within the prime factorization
Multiply the resulting sums
Whats the prime factorization of 100?
2^2 x 5^2
How do you find the GCF of large complex numbers ?
Find the prime factorization of all numbers and stack them in columns.
See what prime factors they have in common
And multiply what they have in common to get the GCF
(In common means the number must be in the row of all the numbers)
How do you find the LCM of large complex numbers ?
Find the prime factorization of all numbers and stack them in columns.
See what is the largest prime factors they have in common
And multiply the largest ones from each column together and thats the LCM
(You will have a number from each column
How do you answer a length problem?
Find the prime factorization of the number and count the number of prime factors it has whether unique or repetitious
What is 2^5 ?
32
What is 2^8 ?
256
If a question ask about the smallest possible integer it is asking about …?
A power of 2
What is 3^4 ?
81
What is 3^5 ?
243
How do you use prime factorization to tell you about divisibility of two numbers ?
In fraction form, if a numerator is divisible by the denominator all the factors of the denominator need to be in the numerator
How should you approach divisibility problems?
Do prime factorization
Remember if you dont know the rest of a prime factorization put in question marks for the unknown
Remember the PFs of different
variables never overlap
But the PFs of the same variables do overlap and thus…
Get rid of overlapping elements
For something to be a factor of something that means ….
It can be divided by said number .
I.e. 5 is a factor of 20
How do you obtain the smallest multiple of any integer that is a perfect square?
Find its PF
Make all exponents of its PF even
Multiply and you have your answer
All the factors of a perfect square must have …. ?
Even exponents
What is important to remember about the PFs of cubes?
They are multiples of 3
What is the sum angle d difference divisibility rule?
If I=an integer, the sum and the difference of any two multiples of I are ALWAYS a multiple of I as well
I.e. 49+14=63 and 49-14=35
What is an evenly spaced number set?
Any set of numbers in which the gaps between the terms are the same
I.e. 12,15,18,21 (gap of 3)
What are consecutive integers ?
A set of integers spaced apart by gaps of 1
I.e. 1,2,3,4
What are consecutive multiples ? Give an example.
Integers that are divisible by the gaps between them ?
I.e. 4,8,12,16 ( all divisible by 4)
How do you algebraically represent consecutive integers ?
x, x+1, x+2, x+3
How do you algebraically represent consecutive even or odd integers ?
x, x+2, x+4, x+6
They have a gap of 2
Whats special about the mean of an evenly spaced number set ?
The mean always equals the median
The average of the first and last terms of an evenly spaced number set is …
Equals the mean and median of the entire set
If there is an odd number of integers in a number set what will the mean be …?
An integer
If there is an even number of integers in a number set what will the mean be …?
Not an integer
Whats an inclusive number set?
A number set where the end numbers of the set are to be included in any consideration
Whats an exclusive number set?
They indicate that the end numbers are to be excluded from any consideration
How do you calculate the number of numbers in an inclusive number set ?
Get the difference between the end numbers of the set and add 1
I.e. 100 to 100 inclusive = 901
From X to Y denotes that the list is …
Inclusive
Between X and Y denotes that the list is …
Exclusive
How do you calculate the number of numbers in an exclusive number set ?
Get the difference between the end numbers of the set and subtract 1
I.e. 199 and 299 exclusive = 99
How do you count consecutive multiples of a set?
Determine if the end numbers are multiples of the number in question
Get the difference between the end numbers
Divide by given multiple
Add 1
*However if the end numbers are not in the set round down to a multiple that is IN THE SET
I.e. how many multiples of 4 are there from 20 to 80
80-20=60. 60/4=15. 15+1= 16
How do you count evens and odds for consecutive multiples
Treat them as multiples of 2
Make sure both end numbers are even for evens and odd for odds
How do you count consecutive multiples of exclusive number sets ?
Figure out the ends of the number set. Go up for the first number and down for the last number.
Get the multiple you need by going up from the front and in from the back
Divide by the multiple, then add 1
i.e. for an exclusive set of 99 and 199 the number of odd numbers in the set you do
100 to 198 for the end numbers Round to 101 and 197 197-101=96 96/2=48 48+1=49
Answer 49
How do you find the sum of any set of evenly spaces integers ?
Number of terms x average =sum
I.e. for 40 and 60 inclusive do
60-40=20 , 20+1=21
Avg of 40 and 60= 50
21*50= 1,050
The sum of an odd number of consecutive integers is ALWAYS …
A multiple of the number of integers
I.e. 21,22,23 their sum will be a multiple of 3
The sum of an EVEN number of consecutive integers is NEVER
A multiple of the number of integers
I.e. 2,3,4,5 their sum will not equal a multiple of 4
Set Z contains 4 consecutive multiples of 4. Is the sum of set Z divisible by 4? ….
Yes (they are not consecutive numbers they are multiples. If they were consecutive numbers the answer would be no)
The product of N evenly spaced terms is always a multiple of N!
I.e. if a,b,c, and d are positive consecutive multiples of 5, is abcd a multiple of 12?
Yes because 4!=24 and 12 is a multiple of 24
What is the proper form for remainders ?
I = D * Q + R
I=dividend
D= Divisor
Q=Quotient
R= Remainder
Whats a quick way to generate an example of a remainder ?
Add The remainder and the dividend
Are remainders ever negative ?
No
How do you solve remainder questions that involve decimals ?
Set the remainder up as a fraction and set it equal to the remainder mentioned in the problem
So do (R/D) = (N/100)
N is the remainder listed in the problem (the decimal ) and D is the divisor
The remainder you are looking for is 100
How do you answer the following type of question
If x and y are integers such that (x/y) =53.14, which of the following values could be the remainder when x is divided by y?
Put the remainder value over 100. Reduce it and set it equal to R over the divisor .
I.e. 14/100 = 7/50
So R/y=7/50
So 50 * R= 7 *y
So answer must be divisible by 7
2n + 1 represents what ?
An odd number
2n represents what?
An even number
Are decimals and fractions considered even or odd ?
No
Odd x odd =
Odd
Even +/- Odd is
Odd
What are the two ways to generate an odd number ?
Even +/- odd
And
Odd * odd
Odd / even always generates …
A fraction
Even / Even =
Even
Odd
Fraction
Even / Odd =
Even
Fraction
Odd / odd =
Odd
Fraction
Odd / Even=
Fraction
If asked a question about consecutive integers being even or odd you should …?
Pick two sets of numbers
One beginning w an odd number
Another beginning with an even number
What is the only even prime number ?
2
Remember to test 2 when you have a question testing your knowledge of prime numbers.
Especially for questions testing even and odd knowledge
What does the following statement mean
“Integer q has q distinct factors, and q >1
Q=2
What does the following statement mean
“The difference of any two distinct positive factors of n is odd”
N=2
What is the addition rule?
The sum of an ODD number of consecutive integers is ALWAYS a multiple of the number of integers
The sum of an EVEN number of consecutive integers is NEVER a multiple of the number of integers
How do you solve questions that ask you to determine remainder of an odd or even dividend?
Test out numbers
Use small numbers, test at least 3 numbers
When dividing if the number being divided is larger that will be your remainder. Provide an example.
2 / 4= remainder 2
1 / 4 = remainder 1
How much time should you spend per question ?
No more than 2 minutes
