Integer Properties Flashcards
Integers
Whole, positive and negative numbers including 0
Factor
If a x b = c
then a and b are factors of c
Divisor
If c/a = b
then a is a divisor of c because it divides evenly
Divisible
If c/a = b
then c is divisible by a because it divides cleanly
Factor Pairs
Find pairs of numbers that have a product of the given number.
Ex: 36
1, 36 2, 18 3, 12 4, 9 6, 6
A total of 9 factors (don’t count 6 twice!)
Negative Integers
It’s technically correct that +4 and -4 are factors/divisors of -12 and that -12 is divisible by +4 and -4 but the gmat won’t typically ask you to know this
Divisibility Rules for 2
All even numbers are divisible by 2
Look at the last digit to see if it’s even!
Divisibility Rule for 5
If the last digit of the number is a 5 or a 0, the number is divisible by 5
Divisibility Rule for 4
If the last two digits in the number are divisible by 4, then the entire number is divisible by 4
Divisibility Rule for 3
Add up all the digits in the number. If the sum is divisible by 3, then the entire number is divisible by 3
Divisibility Rule for 9
Add up all the numbers. If the sum is divisible by 9, the entire number is divisible by 9.
Divisibility Rules for 6
For a number to be divisible by 6, it must be both
1) an even number (look at the last digit)
2) divisible by 3 (add up all the digits and if the sum is divisible by 3, the entire number is divisible by 3)
Important Prime Number Facts
1) 1 is NOT a prime number because it only has one factor…itself.
2) 2 is the ONLY even prime number. All other even numbers are divisible by 2 and are therefore not prime.
Testing whether a number less than 100 is prime
We have to check whether the given number under 100 is divisible by one of the prime Numbers less than 10 (2,3,5,7)
If the given number under 100 is NOT divisible by a prime divisor under 10, the number has to be prime.
Fundamental Theorem of Arithmetic
Every positive integer greater than 1 must be:
1) a prime number
OR
2) expressed as a unique product of prime Numbers
Prime Factorization
The unique product of prime Numbers for a number greater than 1 that is itself not a prime number.
The prime factorization of a number is like the DNA of the number, revealing all of its essential ingredients.
Any factor of q must be composed only of prime factors found in q.
Find the number of factors that integer N has:
4 step process:
1) find the prime factorization of N and write it in terms of the powers of prime factors.
2) create a list of the exponents of prime factors (remember to use 1 for a prime factor that has no exponent!)
3) add 1 to every number on the list to create a new list
4) find the product of the new list. That product is the number of factors N has (including 1 and N itself)
Ex: How many factors does 21,600 have?
1) 21,600 = 216 x 100 = 36 x 6 x 10 x 10 = 6 x 6 x 6 x 10 x 10 = (3x2) x (3x2) x (3x2) x (5x2) x (5x2) = 3^3 x 2^5 x 5^2
2) {3, 5, 2}
3) {4, 6, 3}
4) 4 x 6 x 3 = 72
21,600 has 72 different factors including 1 and 21,600
- To find the odd factors, we’d basically repeat this process but ignore all factors of 2*
1) 3^3 x 2^5 x 5^2
2) {3, 2}
3) {4, 3}
4) 4 x 3 = 12
21,600 has 12 off factors
**we have no direct way to find the even factors of a number. We must first find the total number of factors, then the odd number of factors, and then subtract*
For 21,600
72 total factors
minus 12 odd factors =
60 even factors
*overall, the exponents of each prime factor of N delineate the possibilities for the individual factors of N
Squares of Integers
1) in the prime factorization of a perfect square, each prime factor must have an even exponent
N^2 = N x N = (factors of N)(factors of N)
2) if the prime factorization of some unknown integer has all even exponents, that unknown number must be a perfect square
Ex: k = 2^6 x 3^4 x 5^2
k = (2^3 x 3^2 x 5)(2^3 x 3^2 x 5)
So k = 360^2
3) perfect squares always have an odd number of factors (because one of those factors pairs with itself)
36 = 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6
But we only count the 6 once because it is one factor.
4) the only integers with odd numbers of factors are the perfect squares
Greatest Common Factor
The greatest common factor of two numbers is the largest number on the list of common factors shared by both numbers.
To find the GCF, we start by finding the prime factorization for each number
Ex: Greatest common factor of 360 and 800
360 = 3^2 x 2^3 x 5
800 = 2^5 x 5^2
Next, we look at the highest powers are that they have in common for each factor.
In this example:
2 - 3
3 - 0
5 - 1
Finally we multiply that together! So:
GCF = 2^3 x 3^0 x 5^1
= 8 x 1 x 5
GCF = 40
Least Common Multiple
Least Common Multiple (LCM) = Lowest Common Denominator (LCD)
LCM is the lowest multiple two numbers have in common.
To find the LCM, first find the GCF of the two numbers.
For example, 24 and 32
GCF = 2^3 = 8
Next, write each number in the form: GCF x (another factor)
24 = 8 x 3 32 = 8 x 4
The LCM is the product of these three factors:
LCM = 8 x 3 x 4 LCM = 96
LCM is the same as the LCD, which helps us quickly solve addition and subtraction in fractions.
Ex: 1/10 - 1/35
GCF of 10 and 35 is 5
10: 5 x 2
35: 5 x 7
LCD = 5 x 2 x 7 = 70
7/70 - 2/70 = 5/70 = 1/14
Tips:
1) if A is a factor of R, then the LCM of A and R must be R
- ex: the LCM of 8 and 24 = 24
2) if it’s obvious that A and B have no factors in common greater than 1, their LCM will have to be their product, A x B
GCF-LCM Formula
For any two integers P and Q:
LCM = p x q / GCF
*MAKE SURE you simplify this fraction FIRST otherwise it will take way too long to solve.
Even and Odd Integer Rules and Tricks
1) zero IS an even number (but not positive or negative)
2) evens and odds include both positive and negative numbers
3) evens and odds only pertain to integers - any non-integer is neither even nor odd
4) all even numbers are divisible by 2. Even numbers can be expressed as 2k, where k is any integer. The prime factorization of a positive even integer greater than two absolutely has to include 2.
5) no odd number is divisible by 2. Odd numbers can be expressed as (2k + 1) or (2k - 1), where k is any integer. The prime factorization of a positive odd number will never contain a factor of 2.
6) when looking at odd/even problems on GMAT, the first step should be to determine whether or not the test explicitly states the values are integers. If not, you usually can’t determine the answer because the odd/even rules don’t apply to non-integers.
Adding/Subtracting Evens and Odds
Even +/- Even = Even
Odd +/- Odd = Even
Even +/- Odd = Odd
Odd +/- Even = Odd
Adding and subtracting “likes” = even
Adding and subtracting “dislikes” = odd
Multiplying Evens and Odds
Even x Even = Even
Odd x Odd = Odd
Even x Odd = Even
Odd x Even = Even
As long as there is at least one even factor in a product, the product will be even.
The only way a product will be odd is if every single factor is odd.
Dividing Evens and Odds
No general rules
Even / Even = even or odd or not an integer at all
Odd / Odd = odd or not an integer at all
Even / Odd = even or not an integer at all
Odd / Even = is never going to be an integer (because 2 in denominator with no way to cancel it)
Remainder Equation
If D is dividend and S is divisor and Q is quotient and r is remainder, then:
D/S = Q + r/S
remainder is always greater than or equal to zero and less than the divisor.
Rebuilding the Dividend Equation
If D is the dividend and S is the divisor and Q is the quotient and r is the remainder, then:
D = (S x Q) + r
Important when trying to find the dividend!
