2012/035) Show that if a is any integer, then either 5|a^2 or 5|(a^2-1) or 5|(a^2+1)

a^2 = a^2
a^2 - 1 = (a-1)(a+1)
a^2 + 1 = a^2 - 4 ( note that 1 = - 4 mod 5) = (a+2)(a-2)

5 must divide a or a+ 1 or a-1 or a+ 2 or a- 2 ( 5 consecutive numbers)

if 5 divides a the a^2 is divisible

if 5 divides (a+1) or (a-1) the a^2 - 1 id divisible

else if 5 divides (a+2) or (a-2) then a^2-4 or a^2+ 1 is divisible by 5

so one of them is divisible