Reference no: EM131084279
1: (a) What are the last two digits of 32014?
(b) What are the last two digits of (97)2003?
2: (a) Prove that this sequence contains no squares:
11, 111, 1111, 11111, ...
(b) Prove that if 2n + 1 and 3n + 1 are both perfect squares, then n is divisible by 40.
3: (a) Prove that any subset of 55 numbers chosen from {1, 2, 3, ..., 100} must contain two numbers differing by 9.
(b) Prove that for any set of n integers, some subset of them has sum is divisible by n.
4: For a positive integer n, define the digital sum to be the number you get by summing all its digits, then summing all the digits of that number, and repeating until you get a single digit number. Prove that if you take any pair of twin primes (two consecutive odd numbers that are both prime) other than 3 and 5, then the digital sum of their product is 8.
5: The number d1d2 · · · d9 has nine (not necessarily distinct) decimal digits. The number e1e2 · · · e9 is such that each of the nine 9-digit numbers formed by replacing just one of the digits di in d1d2 · · · d9 by the corresponding digit ei is divisible by 7. The number f1f2· · · f9 is related to e1e2 · · · e9 in the same way: that is, each of the nine 9-digit numbers formed by replacing just one of the digits ei by the corresponding fi is divisible by 7.
Show that, for each i, di - fi is divisible by 7.
6: A classy social security number? Use the digits 1 through 9 once only to form a nine-digit number such that the first (leftmost) 8 digits form a number divisible by 8, the first 7 digits form a number divisible by 7, and so on. How many such numbers are there?
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Use the central limit theorem to estimate
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Describe the current phenomenon of missionary movements
: After reading Winter & Hawthorne, chs. 56-61, describe the current phenomenon of missionary movements from the non-Western world. What are the advantages and difficulties for non-Western missionaries?
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What is the firms breakeven point in units
: Cranberry woods spends an average of $9.30 in labor and $12.60 in materials on every unit it sells. sales commissions and shipping amount to another $3.10. all other cost are fixed and add up to 136,000 per month. what's is cranberry's contribution? ..
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What would the risk-free rate
: Stock Y has a beta of 0.8 and an expected return of 8.2 percent. Stock Z has a beta of 0.9 and an expected return of 8 percent. What would the risk-free rate have to be for the two stocks to be correctly priced relative to each other?
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How many such numbers are there
: A classy social security number? Use the digits 1 through 9 once only to form a nine-digit number such that the first (leftmost) 8 digits form a number divisible by 8, the first 7 digits form a number divisible by 7, and so on. How many such numbe..
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Write on role of shamanism in developed religious traditions
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The result of increasing financial leverage in the firm
: Which of the following actions will have the result of increasing financial leverage in the firm? In each case, assume that all other activity in the firm does not change. a. A 2 for 1 stock split b. A new debt issue c. An increase in the firm’s reta..
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Weighted average of that implied by outstanding debt issues
: Filer Manufacturing has 8 million shares of common stock outstanding. The current share price is $80, and the book value per share is $7. Filer Manufacturing also has two bond issues outstanding. The first bond issue has a face value of $75 million, ..
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Workplace or social context
: In a five to six sentence paragraph provide an example of how to effectively deal with a difficult person in either a workplace or social context.
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