Question

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

Answer 1

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.