Question

Show that one and only one out of n , n+2 or n+4 is divisible by 3 , where n is any
positive integer.

Answer 1

Answer:

case I: if n =3q

n is divisible by 3,

n+2 = 3q+2 is not divisible by 3.

n+4 = 3q+4 = 3(q+1)+1 is not divisible by 3.

case II: if n =3q+1

n = 3q+1 is not divisible by 3.

n+2 = 3q+1+2=3q+3 = 3(q+1) is divisible by 3.

n+4 = 3q+1+4 = 3q+5 = 3(q+1)+2 is not divisible by 3.

case III: if n = 3q+2

n =3q+2 is not divisible by 3.

n+2 = 3q+2+2 =3q+4 = 3(q+1)+1 is not divisible by 3.

n+4 = 3q+2+4 = 3q+6 = 3(q+2) is divisible by 3.

thus one and only one out of n , n+2, n+4 is divisible by 3.