Question

Let n be an integer, with n ≥ 2. Prove by contradiction that if n is not a prime number, then n is divisible by an integer x with 1 < x ≤√n.

[Note: An integer m is divisible by another integer n if there exists a third integer k such that m = nk. This is just a formal way of saying that m is divisible by n if m n is an integer.]

Answer #1

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

Prove by contradiction that:
If n is an integer greater than 2, then for all integers m, n
does not
divide m or n + m ≠ nm.

Problem 2: (i) Let a be an integer. Prove that 2|a if and only
if 2|a3.
(ii) Prove that 3√2 (cube root) is irrational.
Problem 3: Let p and q be prime numbers.
(i) Prove by contradiction that if p+q is prime, then p = 2 or q
= 2
(ii) Prove using the method of subsection 2.2.3 in our book that
if
p+q is prime, then p = 2 or q = 2
Proposition 2.2.3. For all n ∈...

Let n be an integer. Prove that if n is a perfect square (see
below for the definition) then n + 2 is not a perfect square. (Use
contradiction) Definition : An integer n is a perfect square if
there is an integer b such that a = b 2 . Example of perfect
squares are : 1 = (1)2 , 4 = 22 , 9 = 32 , 16, · ·
Use Contradiction proof method

Prove the following statements by contradiction
a) If x∈Z is divisible by both even and odd integer, then x is
even.
b) If A and B are disjoint sets, then A∪B = AΔB.
c) Let R be a relation on a set A. If R = R−1, then R is
symmetric.

Prove that for each positive integer n, (n+1)(n+2)...(2n) is
divisible by 2^n

Definition of Even: An integer n ∈ Z is even if there exists an
integer q ∈ Z such that n = 2q.
Definition of Odd: An integer n ∈ Z is odd if there exists an
integer q ∈ Z such that n = 2q + 1.
Use these definitions to prove the following:
Prove that zero is not odd. (Proof by contradiction)

(a) Let N be an even integer, prove that GCD (N + 2, N) = 2.
(b) What’s the GCD (N + 2, N) if N is an odd integer?

Prove that there is no positive integer n so that 25 < n^2
< 36. Prove this by directly proving the negation.Your proof
must only use integers, inequalities and elementary logic. You may
use that inequalities are preserved by adding a number on both
sides,or by multiplying both sides by a positive number. You cannot
use the square root function. Do not write a proof by
contradiction.

1. Let x be a real number, and x > 1. Prove 1 < sqrt(x)
and sqrt(x) < x.
2. If x is an integer divisible by 4, and y is an integer that
is not, prove x + y is not divisible by 4.

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