The place of “Britain”

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https://www.nps.gov/wori/learn/historyculture/declaration-of-sentiments.htm
Please read the document above.
Please provide a brief summary of the document and then answer the following questions.
Who takes the place of “Britain” in this document?
What do you think the reaction was from the public when this document was published?
Did this document effect any type of change for women during that time period?

Sample Solution

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Division notation, mod arithmetic, inverses Division notation

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Homework 1
Division notation, mod arithmetic, inverses
Division notation
1 Use the Division Algorithm to show that 13 6 | 135.
2 Use the Division Algorithm to determine if 13 | −179. If not then find the remainder as well.
3 Show that if n = pqr with integers p, q, r > 1 then then it has a factor ≤
√3 n.
E.g. n = 847 = 7 · 112
is the product of 3 or more primes. The cube root √3
847 = 9.5 and
indeed 847 has a factor ≤ 9.5.
4 Show that if a | b and 4b | c then
2a | (6b + 7c)
You must work with basic definitions, not a theorem.
5 Methodically determine if 221 is prime or composite. You must use the following approach:
ˆ Find the minimum set of potential factors that need to be checked.
ˆ Check all of them using the Division Algorithm.
ˆ Reach a conclusion. If composite then find also find the prime factorization.
6 Approximately how many primes are less than 1, 000, 000? Less than 1, 001, 000?
Your answer should be based on the Prime Number Theorem.
7 Use your previous two answers to fill in the blank: “Approximately one out of every
odd numbers near 1, 000, 000 are prime.”
8 If you want to check if a 20 digit number is prime then how many potential factors do you
need to test?
Hints: A 20 digit number will be less than 1020. You will use the Prime Number Theorem at
some point. Your answer will be in the ballpark of 500 million.
Continue to next page for questions on Modular Arithmetic.
Copyright ©2021 Ravi Montenegro
1
Modular Arithmetic
9 Find (135 mod 13) using the Division Algorithm.
10 Is 75 ≡ −8 mod 17? Do this in two ways:
(a) Compare 75 mod 17 and −8 mod 17.
(b) Determine if 17 | (75 − (−8)).
11 Find (129 · 173) mod 7 without taking the product 129 × 173.
Your answer must use properties of modular arithmetic. If you write 129 × 173 = 22317 at
any point in the work then you’re just repeating methods of question 9 and will get no credit.
12 Show that if 2a ≡ 2b (mod n), and 10 | n, then a ≡ b (mod 5).
E.g. 102 ≡ 312 (mod 30), and 10 | 30, so 51 ≡ 156 (mod 5).
13 Properties of ordinary arithmetic carry over to the mod-arithmetic setting, except when
division is involved. In this problem you will look at one such situation:
(a) In ordinary arithmetic, if x = y then 2x = 2y.
Show that if a ≡ b (mod n) then 2a ≡ 2b (mod n).
(b) In ordinary arithmetic, if 2x = 2y then x = y.
Why? If 2x = 2y then 2x
2 =
2y
2
and so x = y.
However, in mod-arithmetic the following statement is false:
If 2a ≡ 2b (mod n) then a ≡ b (mod n)
i. Give an example where n is even and 2a ≡ 2b (mod n) but a 6≡ b (mod n).
ii. Give an example where n is even and 2a ≡ 2b (mod n) and yet a ≡ b (mod n).
Hint: Pick any value of a, and then pick any b with a ≡ b (mod n).
iii. What if n is odd? Show that if n is odd and 2a ≡ 2b (mod n) then a ≡ b (mod n).
iv. Use properties of mod-inverses to explain why “mod-division by 2” always works
when n is odd but often fails when n is even.
(c) Do you see a pattern? For what type of integers n will it be true that
If 3a ≡ 3b (mod n) then a ≡ b (mod n)
14 Use brute force to find (27−1 mod 47), i.e. check x ≡ 1 mod 47, x ≡ 2 mod 47 until the
inverse is found.
15 Use your inverse to solve the equation
27x + 5 ≡ 2 mod 47
Copyright ©2021 Ravi Montenegro
2

Sample Solution

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Importance of College Education

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Using your first paragraph assignment on why its important for you to get a college education, you  will write a 1000 word 5-paragraph essay on why its important in general for someone to get a college education.You will need to research the reasons and cite from 5 expert sources

SUMMARY OF KEY RATIOS

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Notes:
Total Revenues = Net Service Revenues + Interest Income

Operating Revenues is the same as Net Service Revenues

Earnings Before Interest and Taxes (EBIT) = Net Income + Interest Expense + Income Tax Expense

Net Fixed Assets is the same as Net Property and Equipment

Ratios Formula Definition
Total (profit) Margin Net Income
(Total Revenues)

Note: Total Revenues = Net Service Revenues + Interest Income
Measures the ability to control expenses (dollars of income per dollar of revenue)
Operating Margin Operating Income
Operating Revenues

Note: Operating Revenues is the same as Net Service Revenues Measures the ability to control operating expenses (dollars of operating income per dollar of operating revenue)
Return on Assets (ROA) Net income
Total Assets
Measures the ability of a business’s assets to generate profits (dollars of profitability per dollar of total assets)
Return on Equity (ROE) Net Income
Total Equity
Measures the ability of a business’s equity financing to generate profits (dollars of profitability per dollar of equity investment)
Current Ratio Current Assets
Current Liabilities
Measures the dollars of current assets per dollar of current liabilities – a rough measure of liquidity
Days Cash on Hand (DCOH) (Cash and cash equivalents + Short-term Investments)
((Cost of Service Revenues + Operating Expenses – Depreciation) / 365)
Measures the number of days that a business can continue to pay its bills without new revenue or financing
Debt Ratio Total Liabilities
Total Assets
Measures the proportion of debt in a business’s total financing
Debt-to-Capitalization Ratio Long-term Debt
(Long-term Debt + Equity)
Measures the proportion of debt in a business’s long-term financing

Times Interest Earned (TIE) Ratio (Earnings Before Interest and Taxes (EBIT))
Interest Expense

Note: Earnings Before Interest and Taxes (EBIT) = Net Income + Interest Expense + Income Tax Expense
Measures the dollar of accounting income available to pay each dollar of interest expense
Cash Flow coverage (CFC) Ratio (EBIT + Lease payments + Depreciation Expense)
((Interest Expense + Lease payments + Debt principal) / ( 1 – Tax rate))
Measures the dollar of cash flow available to pay each dollar of fixed financial charges
Fixed Assets Turnover Ratio (Total Revenues)
Net Fixed Assets

Note: Total Revenues = Net Service Revenues + Interest Income
Note: Net Fixed Assets is the same as Net Property and Equipment

Measures the dollars of revenue per dollar of fixed assets

Total Assets Turnover Ratio (Total Revenues)
Total Assets

Note: Total Revenues = Net Service Revenues + Interest Income
Measures the dollars of revenue per dollar of total assets
Average Collection Period (Days in Patient Accounts Receivable) Net Accounts Receivable
(Net Patient Service Revenue / 365) Measures the average number of days it takes to collect a business’s receivables

DUPONT ANALYSIS:

ROE = Total Margin x Total Asset Turnover x Equity Multiplier

=   Net Income      x   Total Revenue       x   Total Assets
     Total Revenue           Total Assets             Total Equity

Sample Solution

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