n = (a^p)
(b^q)
(c^r)
Prime Factorization of an Integer (n)
(p+1)
(q+1)
(r+1)
What equation do you use to find the "Number of factors of n" after performing prime factorization?
If the last digit is even, the number is divisible by 2
Divisibility Rules - 2
If the sum of the digits is divisible by 3
Divisibility Rules - 3
If the last two digits form a number divisible by 4
Divisibility Rules - 4
If the last digit is a 5 or 0
Divisibility Rules - 5
If the number is divisible by both 3 and 2
Divisibility Rules - 6
Take the last digit, double it, and subtract it from the rest of the number
Divisibility Rules - 7
If the last three digits of the number are divisible by 8
Divisibility Rules - 8
If the sum of the digits is divisible by 9
Divisibility Rules - 9
If the number ends in 0
Divisibility Rules - 10
If you sum every second digit and then subtract all other digits and the answer is 0, or the number is divisible by 11
Divisibility Rules 11
If the number is divisible by both 3 and 4
Divisibility Rules - 12
(n/5) + (n/5^2) +....(n/5^k) = # of Trailing Zeroes, Where 5^k < n
How do you determine the number of trailing zeroes in a factorial?
(n/p) + (n/P^2) + (n/p^3).... until p^x < n
Example Question: How many powers of 900 are in 50! ?
How do you find the number of powers of a prime number p in the n! ?
((a1 + an) / 2)*n where a1 is the first term and an is the last
The sum of elements in any evenly spaced set
((1+n)*n)/2
Sum of n first positive integers in a consecutive set
1) Separate the recurring decimal to fraction
2) Annex denominator with "9" as many times as the length of the recurring number
3) Reduce the fraction to its lowest terms
How do you convert a recurring decimal to a fraction?
1) Write down the number consisting with non-repeating digits and repeating digits
2) Subtract non-repeating number from above
3) Divide 1-2 by the number with 9's and 0's: for every repeating digits, write down a 9, and for every non-repeating digit, wri
How do you convert a mixed-recurring decimal to a fraction?
Last n digits of a product of integers are last n digits of the product of last n digits of these integers. For instance, last two digits of 845
9512
408
613 would be the last 2 digits of 45
12
8
13 = 160 = 60
How do you determine the last two digits of a product?
Percent = (Change / Original) *100
How do you calculate Percent Change?
SImple Interest = Principal
Interest Rate
Time
How do you calculate Simple Interest Rate?
Balance(Final) = Principal
(1+(Interest/C)^time
c where C is the number of times compounding
How do you calculate Compound Interest Rate?
1) Open modulus and set conditions
2) To solve/open a modulus, you need to consider 2 situations to find all roots:
a) Positive (or rather non-negative)
b) Negative
What is the 3-step approach to solving equalities and inequalities with absolute value?
x-1 = 4
How can you rewrite Ix-1I = 4 if (x-1) > 0?
-(x-1) = 4
How can you rewrite Ix-1I = 4 if (x-1) < 0?
y = divisor * quotient + remainder (xq+r)
What is the "Remainder Formula"?
Total = A+B+C - (Sum of 2-group overlaps) + (all three) + Neither
Overlapping Sets Formula for 3 sets (Formula 1)
Total = A+B+C - (Sum of EXACTLY 2-group overlaps) - 2*(All three) + Neither)
Overlapping Sets Formula for 3 sets (Formula 2)
The Vertex
The corner of a triangle. A triangle has 3 of these.
Base
Any one of the 3 sides of a triangle
Altitude
The perpendicular from the base to the opposite vertex of a triangle.
Median (Triangle)
In a triangle, this is the line from a vertex to the midpoint of the opposite side.
A = (hb/2)
Area of a Triangle (formula)
The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides
What is the relationship rule for the sides of a triangle?
A = a^2 ((sqrt3) / 4)
How do you calculate the area of an equilateral triangle?
B = 2(sqrt(L^2-A^2))
How do you calculate the base of an isosceles triangle given the length and altitude?
L = sqrt(A^2 + (B/2)^2)
How do you calculate the length of an isosceles triangle given the base and altitude?
A = h*(a+b)/2
Area of a Trapezoid (formula)
Circumference = Diameter * Pi
Circumference of a Circle (formula)
Area = Pi * Radius^2
Area of a Circle (formula)
Length = 2(sqrt(r^2-d^2))
How do you calculate the length of a chord if you know the radius and the perpendicular distance from the chord to the circle center?
Angle = 90L / (pi*r)
How do you calculate an inscribed angle in a circle?
D = sqrt(dx^2 + dy^2)
AKA
D = sqrt ((x2-x1)^2+(y2-y1)^2)
How do you calculate the distance between two points on a coordinate plane?
m = (y2-y1)/(x2-x1)
Slope of a line (formula)
(m+n)/2 = k
Express the following in algebraic form:
k is halfway between m & n on the number line.
Ip-mI = Ip-nI
Express the following in algebraic form:
The distance between p & m on the number line is the same as the distance between p & n.
p = n/N where n = the # of actual occurrences and N = the number of possible occurrences.
Probability Formula (event WILL occur)
q = (N-n)/N OR 1-p where n = the # of actual occurrences and N = the number of possible occurrences.
Probability Formula (event WILL NOT occur)
Find the product of the probability of both independent events
How to you determine the probability of two independent events occurring?
N = n
(n-1)
(n-2)...2*1 = n!
How can you determine the number of possible arrangements of a set? (Formula)
n! / (k!(n-k)!)
What is the Combination Formula? (Objects combined but not ordered in any particular way)
n! / (n-k)!
What is the permutation formula? (An ordered collection of k objects)
sqrt(3a)
How do you determine the diagonal length of a cube?
sqrt(a^2 + b^2 + c^2)
How do you calculate the diagonal length of a cuboid?
pi
r^2
h
Volume of a cylinder (formula)
2(pi)(r)(h)
Surface area of a cylinder w/o bases
2(pi)(r)(r+h)
Surface area of a cylinder including bases
1/3(pi)(r^2)(h)
Volume of a cone
(pi)(r)(sqrt(r^2+h^2)
Outer surface area of a cone w/o base
(pi)(r)(r+sqrt(r^2+h^2))
Outer surface area of a cone including base
V = 4/3 (pi)(r^3)
Volume of a sphere
SA = 4(pi)(r^2)
Surface area of a sphere