Integers Flashcards
integer
a number that can be written withut a decimal or fractional component
whole numbers
inonnegative integers and 0
properties of 0
0x any number=0 0/any numb=0 any numb / 0= undefined (not 0) square root of 0 is 0 0^2 =0 0 is the only number that is equal to its opposite (0=-0) 0= multiple of all numbers 0 is even number
properties of 1
1^2= 1
1 is odd number
1 is not a prime number
even and odd numbers
even= a number must be divisible by 2 without leaving a remainder units digit= 0,2,4,6,8 can be represented as= 2n odd=if it's not divisible by 2 units digit= 1,3,5,7,9 can be represented as 2n-1 or 2n+1
odd+odd= even+even= even+odd= odd-odd= even-even= odd-even= even-odd=
odd+odd=even even+even=even even+odd=odd odd-odd=even even-even=even odd-even=odd even-odd=odd
eveneven
evenodd
oddeven
oddodd
eveneven=even
evenodd= even
oddeven= even
oddodd= odd
even/odd
odd/odd
even/even
even/odd=even
odd/odd=odd
even/even=even or odd
absolute values
is a distance from zero on a number line (hence both -5 and 5 are on the same distance from 0)
signed numbers
another way to refer to a positive or negative number
+++=
—=
+
-
multiplying or dividing two numb with the same sign
the result (product or quotient) is always positive
multiplying or dividing two numb with the different sign
the result (product or quotient) is always negative
when a nonzero base is raised to an even exponent
the result will always be positive
when a nonzero number is raised to an odd exponent
if the original base is positive the result will be positive
if the original base is negative the result will be negative
for any positive integers x and y , y is a factor of x if and only
if x/y is an integer
1<= y<=x –> the factors or divisors
multiple of a number
x is multiple of y if and only
is the product of that number with any integer
if x/y is an integer
x is a multiple of 5= x/5 is an integer
Prime Numbers
any integer greater than 1 that has no factors other than 1 and itself
2 is the only prime number that is even, all the rest are odd
first 30 prime numbers
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
prime factorization
any composite number (a number that is not prime) can be broken down and expressed as the product of its prime factors
finding the number of factors of a particular number (including both prime and non-prime factors)
- find the prime factorization of a number i.e 2,160 = 2^4 * 3^3 * 5^1
- add 1 to the value of each exponent and then multiply these results =(4+1)(3+1)(1+1)= 40
unique prime factors
distinct/different prime factors a number contains
i.e 12= 223 and has 2 distinct prime factors
the number of unique prime factors in a number does not change when
that number is raised to a positive integer exponent
LCM
smallest positive integer into which all the numbers in set will divide
LCM of 2 and 5= 10
how to find LCM
- prime factorization of each integer
- of any repeated prime factors take the largest one (3^2 and 3^3 choose the 3^3 one) if we are left with two of the same power (3^2 and 3^2) just take that number once
- of what is left take all the non repeated prime factors
- multiply together what you found in the steps 2 and 3
= the result is the lcm
when a set of positive int do not share same p.f. the lcm is calculated by
multiplying those numbers (7,6= 7*6=42)
GCF (greatest common factor)
is the largest number that will divide evenly into all of the numbers in the set ( 8,12,16= 4)
finding gcf
prime factorization of all numbers and then multiply the left column only up to the number that divides the entire set
if the set has no common p.f. the gcf is
1
if we know the lcm and gcf of two positive int (x and y) we know the product of x and y
i.e we have two +ve int x and y whose value we dont know, by knowing gcf and lcm we can multiply them together and get the product
xy= LCM(x,y)GCF(xy) or
lcm(x,y)= (X)(Y)/GCF (x,y) and gcf(x,y)=(x)*(y)/lcm(x,y)
if we know the lcm of the set we can find all
unique prime factors
using lcm to solve repeating pattern questions
i.e. light L flashes every 32 sec and light M flashes every 12 sec. if both lights flash together at 8pm, when will be the next time they’ll flash together?
lcm of 12 and 32= 96 (sec) so it will flash again at 8:01:36pm
divisibility
x/y= z +(no remainder) x is dividend (numerator), y is a divisor (denominator) and z is quotient (the result)
two positive int x and y, x/y will yield an int if x is a multiple of y or if y is a factor of x
means the result will yield an int y is a factor of x y is a divisor of x y divides into x x is multiple of y x is dividend of y x is divisible by y
if x is divisible by y, then x is also divisible by all of the factors of y
100/20 is an int (=5), the factors of 20 are 1,2,4,5,10,20 -> 100/1, 100/2, 100/4, 100/5, 100/10 will be integers
divisibility of exponents with the same base
a^3/a^2= a^3-2= a
if x,a,b=int and x is not 0, in order for a^x/a^y to be an int
x>=y
if z is divisible by both x and y, then z must be divisible by
the lcm of x and y
finding a number of multiples are there between two numbers
i.e. numb of multiples of 4 between 12 and 96 inclusive?
last multiple of x in the range - first multiple in the range / x +1
(96-12/4)+1=22
product of 3 consecutive integer is always divisible by
6
two consecutive integer will never share the same prime factors
what is their gcf
always 1
