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GMAT Math Review > Factors, Multiples, Divisibility and Remainders > Flashcards

Factors, Multiples, Divisibility and Remainders Flashcards

1
Q

What is q and r (quotient and remainder) of an integer divided by a larger integer?

A

q = 0, r = integer. Example, 3/11 = 11(0) + 3 (11 is x, 0 is q or n)

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2
Q

What is a factor or divisor?

A

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

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3
Q

How are factors / divisors and quotients and remainders connected?

A

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.

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4
Q

Quotient of an integer divided by factor and divided by non factor.

A

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)

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5
Q

What is a quotient?

A

A non-negative integer. If quotient of a division between integer and non-factor, quotient is the rounded down non-negative integer

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6
Q

Remainder:

  1. Property (range)
  2. How do you find the remainder of 55/6?
A
  1. 0 ≤ r < x (=number you divide integer by = factor)
  2. 55 = 6n + r
    55 = 6    9 + 1
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7
Q

Consecutive even integers

Consecutive odd integers

A

= -2, 0, 2, 4, 6, 8, 10, ….

= -3, -1, 1, 3, 5, ….

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8
Q

Properties of odd and even numbers

A
  1. 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

  1. 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

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9
Q

What is a prime number

A

Number that has as only factors (=divisors) 1 and itself!

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10
Q
  1. Is 1 a prime number?

2. Why?

A

1.
No

2.
Because only divisible by itself

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11
Q

Opposite of prime number

A

Composite number

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12
Q

Exponent number component names

A

Base and exponent

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13
Q

Definition of a squared number

A

Exponent is number of times base is multiplied by itself

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14
Q

Square base > 1

Square 0 < base < 1

A

base > 1 -> square > base (3^2 = 9, 9 > 3)

0 < base < 1 -> square < base (0.2^2 = 0.04, 0.04 < 0.2)

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15
Q

Square root and cube root definition

A

number which squared (/cubed) equals number in square(/cube) root
x such that x^2 (x^3) = n

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16
Q

Real roots v root from squared / cubed root

A

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!)

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17
Q

real square / cubed root of a negative number

A

No real root. Squared / cubed root is IMAGINARY number

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18
Q

meaning of “third power of 5”

A

5 to power of 3

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19
Q

meaning of “third power of 5”

A

5 to power of 3

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20
Q

Other name for decimal point

A

Period (=point)

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21
Q

one position from point
two positions from point
three positions from point

A

Ones/units - tenths
Tens - hundredths
Hundreds - thousandths
Thousands -

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22
Q
  1. how to convert number in scientific notation to decimal point?
  2. trick for moving x digits to the left when x is more than numbers before period
A
  1. 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

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23
Q
  1. name of terms in a multiplication

2. name of terms in division

A
  1. multiplicands, product

2.
dividend, divisor/factor, quotient (and remainder)

24
Q

divide number by a decimal number, e.g. 689.12/14.4

A

move decimals and proceed with normal division (47,856)

25
Q

Properties of operations!

A
26
Q

Properties of operations!

A
27
Q

definition of algebraic expression

A

combination of
variable (=unknown quantity)
constant
arithmetic operations

28
Q

What is an unknown quantity and what is another name for it?

A

it is a quantity that is not known represented by a letter like x and n, aka a variable

29
Q

What is a constant?

A

A known quantity

30
Q

examples of translation from words into algebraic expressions, solve:

  1. x subtracted by y
  2. difference of x and y
  3. twice x
  4. x divided by y
  5. ratio of x to y
  6. quotient of x and y
  7. x to the 4th power
A

1., 2.
x - y

  1. 2x

4., 5., 6.
x/y

7.
x^y

31
Q

solution of equation

A

set of numbers that solve the equation for each of the variables (=set of assignments of constant values to the equation’s variables)

32
Q

what is a set of assignments of constant values to the equation’s variables?

A

the solution of an equation

33
Q

polynomial expression definition

A

ONLY sum of terms of an equation, which can be multiplied by a coefficient and/or raised to a power

34
Q
  1. definition of term of an algebraic expression

2. two special kinds of terms

A
  1. 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.

  1. constant = all parts of the term are numbers
  2. coefficient = the constant hat multiplies the variable in a term
35
Q

examples of polynomials, which one is NOT a polynomial?
2xy^2+3(x^4+3)
(2x+3y)/6x

A

second one because NOT just a sum of terms

36
Q

what is factorisation?

A

combining like terms or common coefficients in a polynomial or in a fraction”!

37
Q

names for

  • polynomial or first degree
  • polynomial of second degree
  • polynomial of third degree
A
  • linear polynomial
  • quadratic polynomial
  • cubed polynomial
38
Q

what are equivalent equations?

A

equations that have the same solution for all variables

39
Q

particular cases of algebraic equation, define:

  • simultaneous equation (WATCH OUT to number of solutions!!)
  • equations that are the same
A
  • 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!!!

40
Q

what are roots of an algebraic equation?

A

its solutions!!!

41
Q

what does it mean to evaluate an equation?

A

to solve the equation, ie to assign a numerical value to the variables (unknown quantities)

42
Q

what are two methods for solving a simultaneous equation? (e.g. solve 6x+5y=29 and 4x-3y=-6 using those two methods!!!)

A
  • 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

43
Q

definition of linear equation

A

equation with max first degree polynomial on both sides (can have more than 1 unknown!)

44
Q
  1. what are possibilities of solutions in linear equations?

2. how do you recognise which of these possibilities a certain equation belongs to?

A
    • 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)
45
Q

solve x^2+6=0

A

no solution because x+5>0 always

46
Q

two properties of factored equations that ease its evaluation (multiplication and division)

A
  • if xy=0, then x=0 or y=0 or y=x=0

- if x/y=0 <=> x=0 and y≠0 !!!!

47
Q

methods to solve (factor) quadratic equations

A
  • 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
48
Q

different cases of solutions in a quadratic equation

A
  • ∆ > 0 -> 2 solutions (distinct!!!)
  • ∆ < 0 -> 0 solutions
  • ∆ = 0 -> 1 solution (2 solutions but same x) -> + x = -b/2a ( = x of parabola vertex)
49
Q

situation of impossibility of a quadratic function

A

x^2+n ≤ 0 where n is a positive number, because x^2 is always positive (therefore higher than 0)!!!

50
Q

operations in LINEAR inequality

A

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)
51
Q
  1. what is a function?

2. how can you represent it schematically?

A

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)
52
Q
  1. what are
    - domain
    - range
    of a function
  2. what are the values of domain and range?
  3. examples of functions with:
    - free domain
    - naturally restricted domain
A
    • 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
53
Q

general rule of functions / expressions (number of outputs / inputs for each input / output)

A

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)

54
Q
  1. what are formulas (definition)?

2. what can formulas be used for? (2)

A

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
55
Q

unit of measure conversion

A

GMAT Math Review (3 decks)

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