Question

Let
n be a positive integer and let S be a subset of n+1 elements of
the set {1,2,3,...,2n}.Show that

(a) There exist two elements of S that are relatively prime,
and

(b) There exist two elements of S, one of which divides the
other.

Answer #1

Let n be a positive integer and let U be a finite subset of
Mn×n(C) which is closed under multiplication of matrices. Show that
there exists a matrix A in U satisfying tr(A) ∈ {1,...,n}

Let n ≥ 1 be an integer. Use the Pigeonhole Principle to prove
that in any set of n + 1 integers from {1, 2, . . . , 2n}, there
are two elements that are consecutive (i.e., differ by one).

Let a positive integer n be called a super exponential number if
its prime factorization contains at least one prime to a power of
1000 or larger. Prove or disprove the following statement: There
exist two consecutive super exponential numbers.

Let a be prime and b be a positive integer. Prove/disprove, that
if a divides b^2 then a divides b.

Let n be a positive integer. Show that every abelian group of
order n is cyclic if and only if n is not divisible by the square
of any prime.

4) Let F be a finite field. Prove that there exists an integer n
≥ 1, such that n.1F = 0F . Show further that the smallest positive
integer with this property is a prime number.

For all problems on this page, use the following setup:
Let N be a positive integer random variable with PMF of the
form
pN(n)=1/2⋅n⋅2^(−n),n=1,2,….
Once we see the numerical value of N, we then draw a random
variable K whose (conditional) PMF is uniform on the set
{1,2,…,2n}.
Write down an expression for the joint PMF pN,K(n,k).
For n=1,2,… and k=1,2,…,2n:
pN,K(n,k)=

Let n be a positive integer and G a simple graph of 4n vertices,
each with degree 2n. Show that G has an Euler circuit. (Hint: Show
that G is connected by assuming otherwise and look at a small
connected component to derive a contradiction.)

Let
U={1,2, 3, ...,3200}.
Let S be the subset of the numbers in U that are multiples of
4, and let T be the subset of U that are multiples of 9. Since
3200 divided by 4 equals it follows that
n(S)=n({4*1,4*2,...,4*800})=800
(a) Find n(T) using a method similar to the one that showed
that n(S)=800
(b) Find n(S∩T).
(c) Label the number of elements in each region of a two-loop
Venn diagram with the universe U and subsets S...

Let λ be a positive irrational real number. If n is a positive
integer, choose by the Archimedean Property an integer k such that
kλ ≤ n < (k + 1)λ. Let φ(n) = n − kλ. Prove that the set of all
φ(n), n > 0, is dense in the interval [0, λ]. (Hint: Examine the
proof of the density of the rationals in the reals.)

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 7 minutes ago

asked 8 minutes ago

asked 8 minutes ago

asked 26 minutes ago

asked 32 minutes ago

asked 39 minutes ago

asked 40 minutes ago

asked 43 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago