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

a) How many positive integers are divisors of 243,000,000? b)
How many positive integers divide both 243,000,000 and
1,440,000

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

Can 1000 be expressed as the sum of two positive integers, one
of which is divisble by 11 and the other by 17? If yes, then in how
many ways?

Three positive integers (a, b, c) with a<b<c are called a
Pythagorean triple if the sum of the square of a and the square of
b is equal to the square of c. Write a program that prints all
Pythagorean triples (one in a line) with a, b, and c all smaller
than 1000, as well the total number of such triples in the end.
Arrays are not allowed to appear in your code. Hint: user nested
loops (Can you...

Three positive integers (a, b, c) with a<b<c are called a
Pythagorean triple if the sum of the square of a and the square of
b is equal to the square of c. Write a program that prints all
Pythagorean triples (one in a line) with a, b, and c all smaller
than 1000, as well the total number of such triples in the end.
Arrays are not allowed to appear in your code. Hint: user nested
loops (Can you...

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

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

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).

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