Question

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

Answer 1

Consider any two positive integers a and b such that a is greater than b.

Then according to Euclid's division algorithm

a=bq+r where q and r positive integers and 0≤r<b

Let a=n,b=3

n=3q,3q+1,3q+2 .....(i)

n+2=3q+2,3q+3,3q+4 .....(ii)

n+4=3q+4,3q+5,3q+6 .....(iii)

In (i), only n=3q is divisible by 3

In (ii), only n+2=3q+3 is divisible by 3

In (iii), only n+4=3q+6 is divisible by 3

Thus only one of them is divisible by 3 in each case.