# Section 3.3: Problem 9 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Show that there is a representable function
such that for every
,
,
(For example,
and
.)

is the least
such that either
, or
is odd, or
:
does not divide
. Here,
iff for all
and
, if
and
, then
, where
is the divisibility relation. This works, because
iff
or
is odd (in which case the function is undefined, so we just return
), otherwise,
is even,
, and
does not divide
for some large
, moreover, for
,
, and
divides
iff
, i.e.
is the least power such that
does not divide
.