Question

Prove that there exists a nonzero number made up of only 1's and 0's such that it is divisible by n. Please use the pigeonhole principle.

Answer #1

Use Pigeonhole Principle to prove that there is some n ∈ N such
that 101n − 1 is divisible by 19. (Hint: Consider looking at
numbers of the form 101k.)

Exercise1.2.1: Prove that if t > 0 (t∈R),
then there exists an n∈N such that 1/n^2 < t.
Exercise1.2.2: Prove that if t ≥ 0(t∈R), then
there exists an n∈N such that n−1≤ t < n.
Exercise1.2.8: Show that for any two real
numbers x and y such that x < y, there exists an irrational
number s such that x < s < y. Hint: Apply the density of Q to
x/(√2) and y/(√2).

1. Prove that an integer a is divisible by 5 if and only if a2
is divisible by 5.
2. Deduce that 98765432 is not a perfect square. Hint: You can use
any theorem/proposition or whatever was proved in class.
3. Prove that for all integers n,a,b and c, if n | (a−b) and n |
(b−c) then n | (a−c).
4. Prove that for any two consecutive integers, n and n + 1 we
have that gcd(n,n + 1)...

Prove that if a is a transcendental number, then a^n is also
transcendental for all nonzero integers n.

Consider the n×n square Q=[0,n]×[0,n].
Using the pigeonhole theorem prove that, if S is a set of n+1
points contained in Q then there are two distinct points p,q∈S such
that the distance between pand q is at most 2–√.

Let A ={1-1/n | n is a natural number}
Prove that 0 is a lower bound and 1 is an upper
bound: Start by taking x in A. Then x = 1-1/n
for some natural number n. Starting from the fact that 0 <
1/n < 1 do some algebra and arithmetic to get to 0 < 1-1/n
<1.
Prove that lub(A) = 1: Suppose that
r is another upper bound. Then wts that r<= 1.
Suppose not. Then r<1. So 1-r>0....

Suppose that s is a nonzero integer and k is an integer
such that 0 ≤ k < a. Compute
gcd(a, a + k). Prove your assertion.

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).

Irrational Numbers
(a) Prove that for every rational number µ > 0, there exists
an irrational number λ > 0 satisfying λ < µ.
(b) Prove that between every two distinct rational numbers there
is at least one irrational number. (Hint: You may find (a)
useful)

1. Decide whether each statement is true or false. Prove your
answer (i.e. prove that it is true or prove that it is false.)
(a) There exists a nonzero integer α such that α · β is an
integer for every rational number β.
(b) For every rational number β, there exists a nonzero integer
α such that α · β is an integer.

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