We will use the language of the hyperreal numbers to illustrate
how the well-known Kummer congruences involving Bernoulli numbers are the
shadows of congruences of infinite hyperintegers. We will also discuss
Euler's intuitions on divergent series and show how the values of the
Riemann zeta function at negative integers agree with the various values
associated to divergent series. We do this by demonstrating that Ramanujan
summation corresponds, in a certain sense, to taking the "standard part"
of an infinite hyperreal number. This talk will be widely accessible.