Question

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.

Answer #1

6. Consider the
statment. Let n be an integer. n is odd if and
only if 5n + 7 is even.
(a) Prove the forward implication of this statement.
(b) Prove the backwards implication of this statement.
7. Prove the following statement. Let a,b, and
c be integers. If a divides bc and
gcd(a,b) = 1, then a divides c.

(a) Prove or disprove the statement (where n is an integer): If
3n + 2 is even, then n is even.
(b) Prove or disprove the statement: For irrational numbers x
and y, the product xy is irrational.

(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?

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.]

1. Let n be an odd positive integer. Consider a list of n
consecutive integers.
Show that the average is the middle number (that is the number
in the
middle of the list when they are arranged in an increasing
order). What
is the average when n is an even positive integer instead?
2.
Let x1,x2,...,xn be a list of numbers, and let ¯ x be the
average of the list.Which of the following
statements must be true? There might...

Let n be an odd integer.
Prove that 5460 | n25 −n

let n be an odd integer ,prove that 5460 | n^25-n

Prove that if an integer is odd, then its square is also odd.
Use the result to establish that if the square of an integer is
known to be even, the integer must be even

Let n be an integer greater than 2. Prove that every subgroup of
Dn with odd order is cyclic.

Prove that if n is a positive integer greater than 1,
then n! + 1 is odd
Prove that if a, b, c are integers such that a2 + b2 =
c2, then at least one of a, b, or c is even.

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