GRE Math Flashcards
WIN
How many factors does 12 have?
12! (1,-1,2,-2,3,-3,4,-4,5,-5,6,-6,12,-12)
The product of an even integer and an odd integer is…
an even integer!
What digits can a prime number end in? How does this help to remember prime numbers below 100?
1,3,7 or 9 (expect for the only even prime 2). You then only need to double check if the number is divisible by 3 (if the number sum is divisible by 3) or by 7- if not, it’s prime!
Is 1 a prime number?
No! (It’s also not composite)
How many even prime numbers are there?
The only even prime number is 2!
What is a positive number that is not prime?
Composite!
What is the relationship between prime and composite numbers?
All composite numbers can be uniquely expressed as a product of factors that are prime numbers/divisors - this expression is called prime factorization
When is 1 a factor or multiple?
1 is a factor of every integer, 1 is only a multiple of 1 and -1
What is 6 divided by 24 in terms of quotient and remainder?
0 remainder 6
What is -32 divided by 3 in terms of quotient and remainder?
-11 remainder 1 (remainders can be negative but it’s not in this case)
For odd ordered roots (like cubed root) how many roots are there for positive and negative numbers?
One root!
For even ordered roots (like square root) how many roots are there for positive and negative numbers?
Two roots for every positive number and no roots for any negative number!
what is 7,532.418 written as exponents of 10?
7(10^3)+5(10^2)+3(10)+2(10^0)+4(10^-1)+1(10^-2)+8(10^-3)
Are (-3^2) and -3^2 the same?
No! The first is +9 and the second is -9!
What is a^0 and 0^0?
a^0=1 and 0^0 is undefined
What is the square root of a) 0 and b) 16? How about wit the square root symbol?
a) 0, b) 4 and -4! With the square root symbol the answer is only the positive root.
√3√10 equals?
√30
√5/√15 equals?
√5/15 = √1/3
What’s the difference in terminology between a repeating and non-repeating decimal?
repeating=rational and non-repeating= irrational
Is the entire real number line an interval?
Yes!
What is (7√2)√2?
7(√2√2) = (7)(2)= 14
What is the triangle inequality?
|r+s| is less than or equal to |r| + |s|
If s is between 0 and 1, how does s^2 compare to s?
s^2 is LESS than s!
What is the base when computing percent increase and decrease?
increase = base is smaller number, decrease = base is larger number!
How do you find all positive divisors?
Do prime factorization. Each prime to every exponent is a factor and then you need to multiply that prime to every exponent by the other prime(s) to every exponent.
Easy way to know if a number is divisible by 3?
If the sum of the numbers are divisible by 3 (ex. 192 -> 1+9+2=12 so is divisible by 3!)
Easy way to know if a number is divisible by 4?
If the last two digits of your number make a number that is divisible by 4, then the number itself is divisible by 4 (example: 144 - 44 is divisible by 4, so 144 will be divisible by 4)
Easy way to know if a number is divisible by 2^x?
If the last x digits of the number in question make a number that is divisible by 2^x, then said number is divisible by 2^x
Easy way to know if a number is divisible by 3^x?
If the sum of the digits of a number divisible by 3^x, then the number itself is divisible by 3^x
Easy way to know if a number is divisible by 5?
The number must end in 0 or 5
Easy way to find the number of positive divisors
Do prime factorization, then add 1 to both exponents and multiply them together (ex. 144 = 2^43^2->(4+1)(2+1)=53= 15!)
How many primes between 0-100?
25!
How many terms is 8/n+p?
One!!
What is the degree of a polynomial?
Greatest degree (exponent) of its terms (degree 2 is quadratic and degree 3 is cubic)
What is the degree of a constant term?
Zero! (x^0=1)
What is the degree of -7xy^3?
Four! (x^1, y^3 -> 1+3=4)
When is 7x(x+2)/2(x+2) undefined?
When x=-2 (denominator is zero)
Identity of a^2-b^2?
=(a+b)(a-b)
Identity of (a+b)^3?
=a^3 + 3a^2b + 3ab^2 + b^3
Identity of (a-b)^3?
=a^3 - 3a^2b + 3ab^2 - b^3
What is true if x^a=x^b (and x is positive and not 1)
a=b!
What is 2^a in fraction form?
1/(2^-a)
(y^3)(y^-1)
y^2
Two fractions that equal x^a-b?
x^a/x^b = 1/(x^b-a)
What is 0^0?
Undefined
(x^a)(y^a) versus (x^a)(y^b)
(xy)^a and (xy)^a+b
(10z)^3=
(10^3)(z^3)=1,000z^3
What are equivalent equations?
Equations that have the same solutions (ex. for both x+1=2 and 2x+2=4, x=1)
Which one is a linear equation: 1) 2x+1=7x, 2) x+y^2=0, 3) xz=3
Only 1! (2x+1=7x)
What is an identity?
Two equations that are equal for all values
QUADRATIC FORMULA?!
negative b plus or minus the square root of b squared minus 4ac over 2a
What is the square root of zero?
Zero!
The slide method of factoring
Move a to a*c and then remember to divide a from each answer at the end!!
When does an inequality sign flip?
When both sides are multiplied or divided by a negative number
What is the decimal version of increasing x by 14%?
1.14x
PRACTICE SOME RATE QUESTIONS ;)
PRACTICE SOME RATE QUESTIONS ;)
Equation for simple interest
value = initial investment * (1 + (percent * time in years/100))
Equation for compound interest
value = initial investment * (1 + (percent/100))^(number of times compounded * time in years)
What are the order of the four quadrants?
From top right counter clockwise (ex in points- Q1: (4,2); Q2: (-4,2); Q3: (-4,-2); Q4: (4,-2))
What is the point (4,2) reflected about the x-axis?
(4,-2)!
What is the point (4,2) reflected about the y-axis?
(-4,2)!
What is the point (4,2) reflected about the origin?
(-4,-2)!
What are the variables in the equation for a line?
y=mx+b, m is the slope and b is the y-intercept
What is the equation for finding the slope?
(y2-y1)/(x2-x1) (rise over run)
What is the slope and equation for a horizontal line?
Slope is zero and equation is y=b
What is the slope and equation for a vertical line?
Slope is undefined and equation is x=a (where a is the x-intercept)
If the slope of a line is 2 what are is the slope of the perpendicular line?
-1/2 (Negative reciprocal)
How do you solve for the y-intercept? the x-intercept?
Find y-intercept by picking any point, putting it in line equation and solving for b. Find x-intercept by solving for when y=0.
What is the point (4,2) reflected about the line y=x?
(2,4)!
How do you find the the reflection over y=x of a line?
Interchange the x and y and then solve for y.
When is a parabola upwards?
When a (the constant for x^2) is positive!
When is a parabola downwards?
When a (the constant for x^2) is negative!
How do you solve for the vertex of a parabola?
x= NEGATIVE b/2a
What is the equation for a circle?
(x-a)^2 + (y-b)^2 = r^2 The center is point (a,b) and r is the radius
What is the equation for absolute value?
piecewise-defined function, y=x when x is greater than zero, y=-x is less than zero
How does y=x^2 relate to y=sqrt or x?
The are reflections over the line y=x
What is the equation of a parabola opening up to the right (vertex at origin)?
For x≥0, the top half is y=sqrt of x and the bottom half is y= -sqrt of x
How is y=|x|+2 different than y=|x|?
y=|x| is shifted UP by 2 units!
How is y=(x+1)^2 different than y=x^2?
y=x^2 is shifted LEFT by 1 unit!
How is y=(x-4)^2 different than y=x^2?
y=x^2 is shifted RIGHT by 4 units!
How is y=2|x-1| different than y=|x|?
y=|x| is shifted RIGHT by 1 unit and stretched upward (away from x-axis) by a factor of 2
How is y= - x^2/4 different than y=x^2?
The parabola is facing downward and is squished toward the x-axis by a factor of 1/4
(s^7)(t^7)=
(st)^7
(x^10)(y^-1) / (x^-5)(y^5)
x^15 / y^6
Two lines intersect, making 4 angles. A=60,B=120,C=60,D=120. Which are opposite angles?
A&C (both 60) and B&D (both 120)
All polygons in GRE are what kind? What is true of each interior angle?
Convex polygon where each interior angle is less than 180.
How many triangles can you fit in a polygon? What does this mean about the sum of all interior angles?
triangles = Number of sides minus 2. Sum of angles = number of triangles * 180 aka (n-2)*180
What is a regular polygon?
All sides and angles are congruent
What is true of the sides of any triangle?
The length of one side must be less than the sum of the other two
What are the angles and side ratio for an isosceles right triangle?
angles are 45,45,90; side ratio is 1:1:sqrt2
What are the angles and side ratio for half an equilateral triangle?
angles are 30,60,90; side ratio is 1 (base), sqrt of 3 (height), 2 (hypotenuse) (*Note: the hypotenuse is 2 times the base)
What are the three propositions to prove congruent triangles?
SSS (side,side,side are = in both triangles); ASA (angle, side, angle); SAS (side, angle, side)
What is true about side ratios with similar triangles?
1) The ratios of corresponding sides from both triangles are the same between all 3 pairs of sides. 2) The ratios between one side with another in the same triangle is the same for the other triangle.
What is true of angles in a parallelogram?
Opposite angles are congruent.
What is the formula for the area of all parallelograms? Is a trapezoid a parallelogram?
A=bh; NO a trapezoid is not a parallelogram (only one pair of sides are parallel instead of both)
What is the formula for the area of a trapezoid?
A=1/2 (b1+b2)(h)
What is the best fractional approximation of pi?
22/7
How is the measure of an arc and it’s central angle related?
They are equal!
The ratio of the length of the arc to the (a)______ is equal to the ratio of the degree measure of the arc/central angle to (b)______.
(a) Circumference; (b) 360 degrees
The ratio of the area of a sector (slice) to the (a)______ is equal to the ratio of the degree measure of the arc/central angle to (b)______.
(a) Full circle area; (b) 360 degrees
If a circle radius intersects a line at a point on the circle at 90 degrees, what is true of the line?
It is a tangent!
If a line intersects a circle at one point, what is true of it’s relation to the radius that intersects that same point?
The lines are perpendicular!
If the center of a circle is on one of the sides of an inscribed triangle, is is a (1)________. This means that the triangle is a (2)________ triangle.
(1) Diameter of the circle; (2) right triangle!
Two or more circles with the same center are called
Concentric circles
How many edges and vertices does a rectangular solid have?
12 edges and 8 vertices
What is the formula for the volume and surface area of a rectangular solid?
V=lwh; SA= 2(lw+lh+wh)
What is formula for the volume of a cylinder?
Area of the base times the height (i.e. V=pir^2h)
What is formula for the surface area of a cylinder?
Area of both bases plus area of the wrap around rectangle (i.e. 2pir^2 + 2pir*h)
What is a relative frequency?
frequency divided by the total number of data
How do histograms directly relate to probability?
The sum of the areas under the bars equal 100% and the area under one bar over 100% is the probability that that item or group of items will show up.
How would you find the degree of the angle/arc for a slice of a pie chart that represents 7% of the whole?
360*.07 = 25.2 degrees
Can a list of number have more than one mode?
Yes! (If more than one number appear the most time)
How do you find Q1 and Q3 of a number list?
Write the numbers in order, spit it into 2 (removing the median if there is an odd # of #s), find the median of the two smaller groups (**Q2 always = median)
What is Q1-3 in terms of 99 Percentiles?
Q1=P25; Q2=P50; Q3=P75
What is the interquartile range?
Q3-Q1
What is the equation for the weighted mean of the list 2,4,4,5,5,5? What is special about the weights?
1(2)+2(4)+3(5)/(1+2+3) = 25/6; The weights add up to the total # of #s
Which is affected by a very high or low outlier? (a)mean (b)median
A! (the median doesn’t change if you change one number to a very high or low number as long as it’s order in the list stays the same)
HOW DO YOU COMPUTE STANDARD DEVIATION?
- Find the mean; 2. Find difference between mean and each value; 3. Square the differences; 4. Find avg of squared differences; 5. Find the positive square root
How does the ‘sample standard deviation’ differ from ‘(population) standard deviation’? When do you use the former?
Sample means dividing the sum of the square differences by n-1 instead of n. This is preferred for a sample of data taken from a larger populations of data.
What is the process of standardization? Why is it useful?
Subtract the mean from each value in a list and divide by the standard deviation. This provides a measure of position relative to the rest of the data independent of the units or absolute values (i.e. how many standard deviations a value is away from the mean in either direction)
How does an empty set { } relate to all other sets?
An empty set is a subset of every set
What is the difference between a list and a set?
In a set repetition isn’t counted and order doesn’t matter! So the set {1,2,3,2} and set {3,1,2} are the SAME!
What does |S| for the set {1,2,3,2}
|S|=3, which is the number of unique elements in the set
A ∩ B? What is it called when there is none?
The INTERSECTION of sets A & B, the set of all elements that are in BOTH (i.e. the overlap). Sets with no overlap are called disjoint or mutually exclusive.
A ∪ B?
the UNION of sets A & B, the set of all elements that are in EITHER A, B or both.
