Question

prove that one and only one out of x,x+2 and x+4 is divisible by 3

Answer 1

Hey

Here is your answer,

We applied Euclid Division algorithm on n and 3.

a = bq +r on putting a = n and b = 3

n = 3q +r , 0i.e n = 3q -------- (1),n = 3q +1 --------- (2), n = 3q +2 -----------(3)

n = 3q is divisible by 3
or n +2 = 3q +1+2 = 3q +3 also divisible by 3
or n +4 = 3q + 2 +4 = 3q + 6 is also divisible by 3

Hence n, n+2 , n+4 are divisible by 3.

Hope it helps you!

Answer 2

As we already know that any positive integer (here x) is of the form 3m,3m+1 or 3m+2 .
So here,
I case=> let a=x=3m+1 which is not divisible by 3.
II case=>a=x+2=3m+1+2=3m+3=3(m+1) which is divisible by 3 .
III case=>a=x+4=3m+1+4=3m+5 which is not divisible by 3.
hence ,proved .
Thank u, hope it helps!^_^