Question

(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?

Answer #1

(a) If N is a an even integer , N= 2x where x is an integer

We know that GCD(a,b) = GCD(a-b,b) if a>=b Until a or b is 0

So, GCD(N+2,N)= GCD(N+2-N,N)= GCD(2,N) = 2 (Since N is even)

Other Way:

So, N+2= 2x +2 = 2(x+1)

So, GCD(N+2,N ) = GCD( 2(x+1), 2x)

GCD of two consecutive integers is always 1. So, GCD of x and x+1 is 1.

And So, GCD(N+2,N) =

GCD( 2(x+1), 2x)

= 2

(b) GCD(N+2,N) if N is an odd integer is 1.

GCD(N+2,N)

= GCD(N+2-N, N)

= GCD(2,N) = 1 (Since N is a odd integer)

Let n be a positive odd integer, prove gcd(3n, 3n+16) = 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.

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

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.

Prove let n be an integer. Then the following are
equivalent.
1. n is an even integer.
2.n=2a+2 for some integer a
3.n=2b-2 for some integer b
4.n=2c+144 for some integer c
5. n=2d+10 for some integer d

Let n be an even integer. Prove that Dn/Z(Dn) is isomorphic to
D(n/2). Prove this using the First Isomorphism Theorem

Discrete Math
6. Prove that for all positive integer n, there exists an even
positive integer k such that
n < k + 3 ≤ n + 2
. (You can use that facts without proof that even plus even is
even or/and even plus odd is odd.)

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)

Let |a| = n. Prove that <a^i> = <a^j> if and only if
gcd(n,i) = gcd (n,j) and |a^i| = |a^j| if and only if gcd(n,i) =
gcd(n,j).

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