Question

Prove that only one of the numbers n-1,n+1 or n+5 is divisible by 3, where n is any positive integer. Explain.

Answer 1

for any positive integer n =3q, 3q+1, 3q+2 
for n+1
(3q)+1 - not divisible by 3 
(3q+1)+1=3q+2 - not divisible by 3
(3q+2)+1=3q+3 - divisible by 3 

for n-1
(3q)-1 -not divisible by 3
(3q+1)-1=3q -divisible by 3
(3q+2)-1=3q+1 -not divisible by3

​for n+3 
(3q)+3 -divisible by 3
(3q+1)+3 =3q+4 -not divisible by 3
(3q+2)+3=3q+5 -not divisible by 3

Answer 2

let n be the arbitrary number
on dividing by 3 q be the quotient & r be the remainder
n=3q+r where r=0,1,2
if r=0
n-1=3q-1
n+1=3q+1
n+5=3q+5