Question

(a) Let N be an even integer, prove that GCD (N + 2, N) = 2....

(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?

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Homework Answers

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)

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