Number Properties Flashcards
How to know if integer is even
Must be divisible by 2 without a remainder
489,987,204 - even or odd
even, 4 in digits
Even integer - digits
all even numbers have an even units digit (0,2,4,6,8)
Odd integer - digits
all odd numbers have an odd units digit (1,3,5,7,9)
Consecutive integers - even/odd pattern
Alternate odd and even (even: 2n, odd: 2n + 1 or 2n - 1)
Odd integer (as defined by 2n)
2n + 1 or 2n - 1
Addition: Even result
Odd + Odd = even
Even + Even = even
Addition: Odd result
Odd + Even = odd
Even + Odd = odd
Subtraction: Even result
Odd - Odd = even
Even - even = even
Subtraction: odd result
odd - even = odd
even - odd = odd
Multiplication: even result
Any even number times any integer
Even x even = even
Even x odd = even
Odd x even = even
347,867 x 408 = odd or even
even
Multiplication: odd result
Odd x odd = odd
39 x 65 x 3 = odd or even
odd, product of a set of two or more integers is odd if every number in the set is odd
Division: even result
even number divided by an odd number *ONLY works when an integer divides evenly into the other
*Even number divided by an even number can be odd or even
Division: odd result
odd number divided by odd number *ONLY works when an integer divides evenly into the other
12/4 vs 12/3 = even or odd
Odd vs. even, even divided by even can be odd or even but even divided by odd is always odd (if divide evenly)
Odd number divided by even number
N/A - odd number cannot be divisible by even number (i.e. 5/2 or 21/4)
Even number divided by 2
Remainder 0, always divisible by 2
Odd number divided by 2
Remainder of 1 (3/2, 5/2, 11/,2)
8xy/2 is an integer
Must be even
Multiplying two numbers with the same sign
Always positive (pos x pos = pos, neg x neg = pos)
-23 x -4 = pos or neg
Pos
Dividing two numbers w the same sign
always positive (neg/neg = pos, pos/pos = pos)
Multiplying two numbers with diff signs
Always neg (pos x neg = neg, neg x pos = neg)
Dividing two numbers with diff signs
Always neg
XY = pos or neg
Pos
-X/-Y = pos or neg
Pos
-Y/X = pos or neg
Neg
M^5 < 0 tells you
M is negative
value raised to odd exponent < 0
Base must be negative number
value raised to even exponent < or > 0
Always >0 because even power is a positive number regardless of whether base is - or +
When is y a factor of x (both pos. integers)
if y/x = integer
Factor inequality rule
if y is a factor of x then 1
Multiple/Factor =
Multiple over factor = integer
How many factors of a # exist
Finite number, an integer only can be divided evenly by a set number of other integers
How many multiples of a # exist
Infinite, can multiply a number by any number 0 - infinity
X is a multiple of Y if
x/y = integer
Prime Factorization
Divide the number by the first prime number 2 (or 3, 5, 7) until the only numbers left are prime numbers. Write the number as a product of prime numbers.
Total number of Factors
Number N = (p^a)(q^b)(r^c)
Where, p, q and r are prime factors of the number n.
a, b and c are non-negative powers/ exponents
Number of factors of N = (a+1)(b+1)(c+1)
Product of Factors
Product of factors of N = N ^ No. of factors/2
Unique prime factors of x^2 vs x^6
The same, # of unique prime factors remains constant even when that number is raised to a positive power
Least Common Multiple
The smallest positive integer into which all of the numbers in the set will divide
- Find the prime factorization of each integer
- Pull out unique numbers (if number is shared, pull out the largest exponent)
- Multiply these unique numbers
X & Y share no prime factors, what is their LCM
XY
Greatest Common Factor
The largest number that will divide evenly into all numbers in a set
- Find the prime factorization of each integer
- Pull out the repeated prime numbers with the smallest exponent
- Multiply these repeated #s
*If no repeated prime factors are found, the GCF is 1
LCM & GCF of Y if Y divides evenly into X
LCM is X & Y is X and GCF of X & Y is Y
Product of XY if you have LCM & GCF
LCM x GCF
XY =
(LCM of X&Y) x (GCF of X&Y)
LCM (x,y) =
XY/GCF of x & y
GCF (x,y) =
XY/LCM of x & y
How to find the unique prime factors of a set
The LCM will provide all unique prime factors of the set (x,y,z) and the unique prime factors of the product of the numbers in the set (xyz)
Is z a factor of x
If Y is a factor of X and Z is a factor of Y, then Z is a factor of X
Remainder Equation
X/Y = Q + R/Y where Q is the integer and R is the non-negative remainder
Finding number of digits
- Prime factorize the number(s)
- Count the number of 5x2 pairs (each = 1 trailing 0)
- Multiply all unpaired 5’s, 2’s, or any other nonzero prime factors and count those digits
- Add up the # of trailing zeros + # of digits in the product of all non-paired 5x2s
Leading zeros
Zeros that occur to the right of the decimal point but before the first non-zero number
0.2 has no leading zeros, 0.02 has one leading zero, 0.002 has two leading zeros
Calculating leading zeros
If x is a number with k digits, then 1/x will have k-1 leading zeros
*UNLESS x is a perfect power of 10 (i.e. 10, 100, 1000), in which case there will be k-2 leading zeros
Factorials (n!)
multiplication of all integers from 1 - n, inclusive for any positive integer n
1!
= 1
0!
= 1
Rules of Product of consecutive integers
Always divisible by any of the integers in the sett
Divisible by any of the factor combinations of the numbers
Product of any set of n consecutive positive integers is divisible by n!
What’s the largest number that must be a factor of any four consecutive integers
4! = 4x3x2x1 = 24, 24 is the largest
Process: Finding the number of times prime number x divides into factorial y!
- Divide y by increasing values of x raised to a power until you get to 0
- Add up the whole numbers from the division and that is the number of times x will be in factorial y!
How many times does 3 appear in 21!
21/3 = 7 21/3^2 = 2 (ignore remainder) 21/3^3 = 21/27, 0
Total number of times 3 appears in 21 is 9
Process: Finding the number of times NON-prime number x divides into factorial y!
- Break x into prime factors (keep exponents the same)
- Use the larger prime number in the shortcut (divide y by increasing powers of that number), and add up the whole numbers
Process: Finding the number of times NON-prime number x divides into factorial y! if x can be re-written as p^k
- Apply the factorial divisibility shortcut to determine the quantity of p in y! (i.e. the simplified base)
- Set the exponents as less than or equal to and simplify (i.e. 2^3n
Dividing a whole number by 10
The remainder will be the units digit of the numerator
Evenly spaced set
The numbers in the set increase by the same amount and therefore share a common difference (11, 22, 33 or 1,2,3)
Could be 1. Set of consecutive integers (including even and odd)
- Consecutive multiples of a given number (2,4,6,8 or 10,20,30,40)
- Set of consecutive numbers with a given remainder when divided by a certain integer (1,6,11,16,21 or 3,7,11,15,19)
Rule of 2 consecutive integers & prime factors
Two consecutive integers will never share the same prime factors, so the GCF of 2 consecutive integers is 1
GCF(n,n+1) = 1
How to find numbers inclusive
To find the number of integers btwn 2 numbers inclusively, subtract the two “endpoint” numbers and add 1
How many integers are between 2 and 12 inclusive
11, 12-2+1
