Factors, Multiples, Divisibility and Remainders Flashcards
What is q and r (quotient and remainder) of an integer divided by a larger integer?
q = 0, r = integer. Example, 3/11 = 11(0) + 3 (11 is x, 0 is q or n)
What is a factor or divisor?
Factor or divisor is a number that is part of another number. y = factor*n (where n is the quotient)
E.g., 28 = 8n (8 is a factor of 28).
For a factor / divisor, there is no remainder in a division, so r = 0: 28 = 8q + 0
How are factors / divisors and quotients and remainders connected?
A factor is the number an integer is divided by if the division has no remainder. Otherwise, the number the integer is divided by is just x. Result (quotient) is q or n. n if the number is a factor of integer, q otherwise.
Quotient of an integer divided by factor and divided by non factor.
Divided by factor: q = n and r = 0 (e.g., 33/11, q=3, r=0)
Divided by non-factor: q = n but r ≠ 0 (e.g. 33/7, q= 4, r=5)
What is a quotient?
A non-negative integer. If quotient of a division between integer and non-factor, quotient is the rounded down non-negative integer
Remainder:
- Property (range)
- How do you find the remainder of 55/6?
- 0 ≤ r < x (=number you divide integer by = factor)
- 55 = 6n + r
55 = 69 + 1
Consecutive even integers
Consecutive odd integers
= -2, 0, 2, 4, 6, 8, 10, ….
= -3, -1, 1, 3, 5, ….
Properties of odd and even numbers
- Every number that has 2 as a factor (= multiplied by any even number) or product btw any even number and other number is EVEN
Otherwise product is ODD
= EVENanything = EVEN
ODDODD = ODD
- Sum / subtraction of odd/odd and even/even numbers is EVEN
Otherwise, sum / subtractions are ODD
EVEN+EVEN / EVEN-EVEN= EVEN
ODD+ODD / ODD-ODD = EVEN
EVEN+ODD / EVEN-ODD = ODD
What is a prime number
Number that has as only factors (=divisors) 1 and itself!
- Is 1 a prime number?
2. Why?
1.
No
2.
Because only divisible by itself
Opposite of prime number
Composite number
Exponent number component names
Base and exponent
Definition of a squared number
Exponent is number of times base is multiplied by itself
Square base > 1
Square 0 < base < 1
base > 1 -> square > base (3^2 = 9, 9 > 3)
0 < base < 1 -> square < base (0.2^2 = 0.04, 0.04 < 0.2)
Square root and cube root definition
number which squared (/cubed) equals number in square(/cube) root
x such that x^2 (x^3) = n
Real roots v root from squared / cubed root
Positive has 2 real square roots: positive and negative x
Positive number has 1 real cubed root: positive x
When you denote √ x^2, result is only positive x (so, √ x^2 = |x|, otherwise x could be either positive or negative!)
real square / cubed root of a negative number
No real root. Squared / cubed root is IMAGINARY number
meaning of “third power of 5”
5 to power of 3
meaning of “third power of 5”
5 to power of 3
Other name for decimal point
Period (=point)
one position from point
two positions from point
three positions from point
Ones/units - tenths
Tens - hundredths
Hundreds - thousandths
Thousands -
- how to convert number in scientific notation to decimal point?
- trick for moving x digits to the left when x is more than numbers before period
- move period x digits to right if positive exponent
move period x digits to left if negative exponent
2.
add zeros of -x-(numbers before period)
e.g., 2.3 * 10^-4 has 4-1 = 3 zeros
156.6 * 10^-4 has 4-3 = 1 zero
- name of terms in a multiplication
2. name of terms in division
- multiplicands, product
2.
dividend, divisor/factor, quotient (and remainder)
divide number by a decimal number, e.g. 689.12/14.4
move decimals and proceed with normal division (47,856)
Properties of operations!
Properties of operations!
definition of algebraic expression
combination of
variable (=unknown quantity)
constant
arithmetic operations
What is an unknown quantity and what is another name for it?
it is a quantity that is not known represented by a letter like x and n, aka a variable
What is a constant?
A known quantity
examples of translation from words into algebraic expressions, solve:
- x subtracted by y
- difference of x and y
- twice x
- x divided by y
- ratio of x to y
- quotient of x and y
- x to the 4th power
1., 2.
x - y
- 2x
4., 5., 6.
x/y
7.
x^y
solution of equation
set of numbers that solve the equation for each of the variables (=set of assignments of constant values to the equation’s variables)
what is a set of assignments of constant values to the equation’s variables?
the solution of an equation
polynomial expression definition
ONLY sum of terms of an equation, which can be multiplied by a coefficient and/or raised to a power
- definition of term of an algebraic expression
2. two special kinds of terms
- a part of an equation that can include variables and coefficients and can be a sum or a subtraction (=either a constant or a variable or the product between constants and/or variables)
2.
- constant = all parts of the term are numbers
- coefficient = the constant hat multiplies the variable in a term
examples of polynomials, which one is NOT a polynomial?
2xy^2+3(x^4+3)
(2x+3y)/6x
second one because NOT just a sum of terms
what is factorisation?
combining like terms or common coefficients in a polynomial or in a fraction”!
names for
- polynomial or first degree
- polynomial of second degree
- polynomial of third degree
- linear polynomial
- quadratic polynomial
- cubed polynomial
what are equivalent equations?
equations that have the same solution for all variables
particular cases of algebraic equation, define:
- simultaneous equation (WATCH OUT to number of solutions!!)
- equations that are the same
- equations that must be solved together: the solution/s for the variable/s must satisfy ALL the equations at the same time.
might be more than two solutions for each variable!!!!!
- equations that are different but yield the same solution for a given variable/s, so they can be considered equivalent!
certain equations might have infinite solutions!!!
what are roots of an algebraic equation?
its solutions!!!
what does it mean to evaluate an equation?
to solve the equation, ie to assign a numerical value to the variables (unknown quantities)
what are two methods for solving a simultaneous equation? (e.g. solve 6x+5y=29 and 4x-3y=-6 using those two methods!!!)
- writing one expression in terms of the other
- getting same coefficient for one value and then subtracting equation os that it only has one variable
definition of linear equation
equation with max first degree polynomial on both sides (can have more than 1 unknown!)
- what are possibilities of solutions in linear equations?
2. how do you recognise which of these possibilities a certain equation belongs to?
- one solution for each variable
- infinite solutions for each variable
- zero solutions
- when solving equation gets no contradiction or no trivial equation
- when solving get a trivial equation!!! (e.g., 0 = 0)
- when solving get a contradiction (e.g., 4=5 or two equations with same coefficients and different result = 3x+4y = 17 -> 6x+8y=34 and 6x+8y=35)
solve x^2+6=0
no solution because x+5>0 always
two properties of factored equations that ease its evaluation (multiplication and division)
- if xy=0, then x=0 or y=0 or y=x=0
- if x/y=0 <=> x=0 and y≠0 !!!!
methods to solve (factor) quadratic equations
- factorisation of a (like term=) common factor !
- trinomio particolare (2 types!)
- a^2-b^2
- square = a^2 - 2ab - b^2 or a^2 - 2ab + b^2
- formula for x
different cases of solutions in a quadratic equation
- ∆ > 0 -> 2 solutions (distinct!!!)
- ∆ < 0 -> 0 solutions
- ∆ = 0 -> 1 solution (2 solutions but same x) -> + x = -b/2a ( = x of parabola vertex)
situation of impossibility of a quadratic function
x^2+n ≤ 0 where n is a positive number, because x^2 is always positive (therefore higher than 0)!!!
operations in LINEAR inequality
addition / subtraction:
same procedure as for inequality (same effects on numbers as equality!!!)
multiplication / division:
- by a positive number same procedure as for inequality (same effects on numbers as equality!!!)
- be a negative number revert the sign!!! (flip it around)
- what is a function?
2. how can you represent it schematically?
1.
short way of writing the value a variable takes when another variable takes a specific value (short because it hides algebraic expression)
2. input -> expression (hidden!) -> output x = input expression = f output = f(x)
- what are
- domain
- range
of a function - what are the values of domain and range?
- examples of functions with:
- free domain
- naturally restricted domain
- all values that inputs of a function can get (sometimes might be restricted arbitrarily / manually)
- all values that outputs of a function can get (derive from domain, automatic / natural)
2.
domain usually assumed to be all values such that output is then a real number
- free domain = polynomial function! (that contains no square roots)
- naturally restricted domain = ratio (with variable in denominator), square root of a variable, absolute value function!
- naturally restricted range = parabola / even exponents
general rule of functions / expressions (number of outputs / inputs for each input / output)
each input always has one and only one output, but outputs be the same, i.e., one output might have different possible outputs (no vertical line but horizontal line possible)
- what are formulas (definition)?
2. what can formulas be used for? (2)
1.
formulas algebraic expressions with variables that have a meaning associated with it!
2.
- for all meanings of problems (word problems, physics questions, biology relationships, math and geometry relations, …)
- to convert units of measure
unit of measure conversion
