Question

show that one and only one out of n,( n+ 1) and( n + 2 )is divisible by 3 Where n is any positive integer

Answer 1

by Euclid's division lemma n=bq+r where n is some integer and put b=3 then r=0,1,2 because 0≤r<3

so, put n=3q
n=3q+1
n=3q+2

when n=3q
n+1=3q+1
n+2=3q+2
when n=3q+1
n+1=3q+1+1=3q+2
n+2=3q+1+2=3q+3
when n=3q+2
n+1=3q+2+1=3q+3
n+2=3q+2+2=3q+3

so, when we put n=3q then n is divisible by 3, when we put n=3q+1 then n+2 I'd divisible by 3 and when we put n=3q+2 then n+1 I'd divisible by 3

hope it helps!!