Question

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).

Answer #1

Prove the following: Let n∈Z. Then n2 is odd if and
only if n is odd.

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.

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.

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 odd integer.
Prove that 5460 | n25 −n

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.

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

Let n be any integer, prove the following statement:
n3+ 1 is even if and only if n is odd.

If n is an odd integer, prove that 12 divides
n2+(n+2)2+(n+4)2+1. Please provide
full solution!

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.

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