Question

a) How many positive divisors does 144 have?

b) What is the sum of the positive divisors of 144?

c) Which positive integers have an odd number of positive divisors? (Prove your answer)

Answer #1

How many ways are there to represent a positive integer n as a
sum of (a) k non-negative integers? (b) k positive integers? Note:
the order of summation matters. For example, take n = 3, k = 2.
Then the possible sums in (a) are 3+0, 2+1, 1+2, 0+3

how many positive integers less than 1000 have no
repeated digits?

For which positive integers n ≥ 1 does 2n > n2 hold? Prove
your claim by induction.

3. Consider the SF4 molecule.
b) How many stretching vibrations does the molecule have, and what
are their symmetries? Which ones appear in an IR spectrum? Which
ones appear in a Raman spectrum? Sketch the stretching modes.
c) How many bending vibrations does the molecule have, and what are
their symmetries? Which ones appear in an IR spectrum? Which ones
appear in a Raman spectrum?
Dont forget to sketch the stretching nodes

For each of the statements below, say what method of proof you
should use to prove them. Then say how the proof starts and how it
ends. Pretend bonus points for filling in the middle.
a. There are no integers x and y such that x is a prime greater
than 5 and x = 6y + 3.
b. For all integers n , if n is a multiple of 3, then n can be
written as the sum of...

Counting theory: Find how many 4-digit positive
integers are there with no repeating digits (e.g.: 5823) or where
digit repetition is allowed but all digits must be odd (e.g.: 5531
satisfies this condition but 7726 and 6695 do not since they
contain even digits).

A. Find gcd(213486, 5423) by applying Euclid’s algorithm. B.
Estimate approximately how many times faster it will be to find gcd
(213486, 5423) with the help of the Euclid's algorithm compared
with the alroithm based on checking consecutive integers from min
{m,n} down to gcd(m,n). You may only count the number of modulus
divisions of the largest integer by different divisors.

1.)
a.)How many kinetic energy modes does a triatomic molecule
have?
b.)Treating the molecule as one particle, how many translational
kinetic energy terms does it have? what about rotational kinetic
energy terms?
c.)How many vibrational kinetic energy modes can this molecule
have, if the total number of kinetic energy modes is what you found
in a.
d.)Count all vibrational modes. Categorize the modes into types
of kinetic and potential energy modes
e.)Find the total number of modes for a gas...

A. Define a positive externality.
B. What problem does the provision of a good/service with a
positive externality pose for free markets? Use a graph to
strengthen your answer.
C. Give two examples of positive externalities and explain how
they work.
D. How does government encourage solutions to positive
externalities?List all three policy choice given in the
textbook.
Please use text it's easier to understand.

1.) How many “words” are there of length 4, with distinct
letters, from the letters {a, b, c, d, e, f}, in which the letters
appear in increasing order alphabetically. A word is any ordering
of the six letters, not necessarily an English word.
2.) Prove that every graph has an even number of odd nodes.

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