GCD/LCM
Define GCD and LCM.
GCD: Greatest common divisor. GCD(A,B) is the largest number which divides both A and B.
LCM: Least (lowest) common multiple. LCM(A,B) is the smallest number which A and B both divide.
True or false, the prime factorization of 68 is 21 x 341
False. It is 22 x 171
Name three types of numbers and describe them in terms of sets
natural numbers N = {1, 2, 3, ...}
fractions Q = {p/q : p, q ∈ N, q ≠ 0}
integers Z = {... -2, -1, 0, 1, 2, ...}
Complete the sentence:
When turning a fraction to a decimal, we ___________
use long division
Describe the difference between simple and compound interest
Simple interest accrues based only on r% of the principal amount for t years. Compound interest accrues based upon the principal amount AND the year's worth of interest at a rate of r%. Interest earns interest.
Solve for LCM(2, 7, 10)
LCM(2, 7, 10) = 21 x 51 x 71 = 70
Define the Fundamental Theorem of Arithmetic
Every number can be written as a product of prime numbers in exactly one way
The set of all irrational numbers includes...
the set of all non-repeating decimals (ie, 𝝅 )
Complete the sentence:
When turning a decimal to a fraction _______
we multiply by tens twice and subtract
Solve for A:
$1,500 invested at an annual simple interest rate of 3% for 12 years.
Formula: A = P(1+rt)
A(12) = 1500 (1 + 0.03 x 12)
A(12) = 1500(1.36)
= $2040
Three horses entered a race around a track. The brown horse takes 11 minutes to run one lap. The black horse takes 16 minutes to run one lap and it takes the spotted horse 12 minutes to run one lap. If all three begin the race at the same time, how many minutes will it take for all three to be at the starting point again?
LCM(11, 12, 16) = 24 x 31 x 111
Complete the prime factorization for 322.
DO NOT USE A CALCULATOR
21 x 71 x 231
The set of all rational numbers includes...
The set of all fractions Q aka the set of all repeating decimals (also natural numbers N and integers Z)
Convert 5.4444... into a fraction
100x = 544.444...
10x = 54.444...
90x = 490
x = 490/90
You got a bonus of $7,500 and you want to start a college fund for your child. You find an account paying 3.14% annual interest compounded every 2 months. If your child just turned two years old, how much will you have when they turn 18?
P = 7500; r = 0.0314; t = 16 years; n = 6 times per year
A(16) = 7500(1+0.0314/6)6 x 16 = $12,374.95
There are 40 girls and 32 boys who want to play an intramural sport. If each team must have the same number of girls and the same number of boys, (1) what is the greatest number of teams that can participate? (2) How many girls and boys will be on each team?
(1) GCD(40,32) = 23 x 50 = 8 teams
(2)
23 x 51 / 23 x 50 = 5 girls per team
25 x 50 / 23 x 50 = 4 boys per team
Is 607 a prime number? Explain/show work to justify your answer.
DO NOT USE A CALCULATOR
23 x 23 = 529
29 x 29 = 841 (too large)
2 -> No
3 -> No
5 -> No
7 -> No
11 -> No
13 -> No
19 -> No
23 -> No
So YES, 607 is prime
define "infinite decimal expansion" and provide an example to justify your definition
infinite decimal expansion means that any decimal can be defined to any FINITE number of decimal places (in order to use it algebraically).
Ex: pi is not 3.14159..., it is "pi to one hundred decimal places"
Convert 19/27 into a repeating decimal
0.703703...
How much would you need to deposit in an investment fund if you'd like to have $40,000 after 11 years? The investment account earns 2.4% annual compound interest, compounded every 3 months.
40,000 = P (1 + 0.024/4)4 x 11
40,000 = P(1.006)44
40000 = P(1.301)
P = $30,745.58
Kiara baked 60 oatmeal cookies and 60 chocolate chip cookies for a bake sale. She left them cooling in the kitchen, and when she returned, her dog had eaten 8 oatmeal cookies! With no time to bake more, she has to figure out the greatest number of trays she can make with an equal amount of oatmeal and chocolate chip cookies. (1) How many trays can she make? (2) How many of each cookie will be on each tray?
(1) GCD(60, 52) = 22 x 30 x 50 x 130 = 4 trays
(2)
22 x 31 x 51 x 130 / 22 x 30 x 50 x 130 = 31 x 51 = 15 chocolate chip
22 x 30 x 50 x 131 / 22 x 30 x 50 x 130 = 131 = 13 oatmeal cookies
Complete a prime factor tree diagram for 300
Find the number of divisors of 300 and list at least 4 divisors using the tree diagram.
DO NOT USE A CALCULATOR
300 = 22 x 31 x 52
(2+1) x (1+1) x (2+1) = 18 divisors
Describe the set of non-repeating decimals in symbols using principles of set theory.
Hint: Consider the diagram of real numbers R from the notes
Non-repeating decimals = R \ Q
Convert 18.435656...into a fraction
920861/49950
You are considering where to invest your inheritance money of $25,000. One account earns 7.25% interest compounded quarterly, and another will earn 7.15% compounded every month. Which is the better account to invest into for 15 years? Explain your reasoning.
A(15) = 25000(1 + 0.0725/4)4 x 15 = $73,452 after 15 years
A(15) = 25000(1 + 0.0715/12)12 x 15 = $72,078 after 15 years