Question

Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer.

Answer 1

Let n,n+4,n+8,n+12,n+16 be integers.
where n can take the form 5q, 5q+1 ,5q+2 , 5q + 3 , 5q + 4.
Case I When n=5q
Then n is divisible by 5.
but neither of 5q+1 ,5q+2 , 5q + 3 , 5q + 4 is divisible by 5.
Case II when n=5q+1
Then n is not divisible by 5.
n+4 = 5q+1+4 = 5q+5=5(q +1),
which is divisible by 5.(else not)
Case III when n=5q+2
Then n is not divisible by 5.
n+8 = 5q+2+8 =5q+10=5(q+2),
which is divisible by 5.(else not)
Case IV when n=5q+3
Then n is not divisible by 5.
n+12 = 5q+3+12 =5q+15=5(q+3),
which is divisible by 5.(else not)
Case V when n=5q+4
Then n is not divisible by 5.
n+16 = 5q+4+16 =5q+20=5(q+4),
which is divisible by 5.(else not)
Hence, one of n, n+4,n+8,n +12 and n+16 is divisible by 5.