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.

1)Let ? be an integer. Prove that ?^2 is even if and only if ?
is even. (hint: to prove that ?⇔? is true, you may instead prove ?:
?⇒? and ?: ? ⇒ ? are true.)
2) Determine the truth value for each of the following
statements where x and y are integers. State why it is true or
false. ∃x ∀y x+y is odd.

(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.]

Let p be an odd prime and let a be an odd integer with p not
divisible by a. Suppose that p = 4a + n2 for some
integer n. Prove that the Legendre symbol (a/p) equals 1.

1. Let n be an integer. Prove that n2 + 4n is odd if and only if
n is odd? PROVE
2. Use a table to express the value of the Boolean function x(z
+ yz).

Prove that 1+2+3+...+ n is divisible by n if n is odd. Always
true that 1+2+3+...+ n is divisible by n+1 if n is even? Provide a
proof.

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

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