The function:
Definition:
Given a real number x, the expression represents the largest integer
. Also, we denote by
the closest integer to x:
Theorem:
Let x and y be real numbers. Then,
(i) , if m is an integer;
(ii) ;
(iii) the number of positive integers and division by a positive integer a is equal to
.
Theorem (Legendre’s Theorem):
Let p be a prime number and n a positive integer. Then, the largest exponent such that
is given by
,
this last sum being in fact a finite sum since when
. It follows that
Some classic questions based on integer parts and their solutions:
(1) Let . Show that
(1a)
(1b)
(1c)
(1d)
(1e)
(2) Show that is an even integer for each
.
(3) Let . Show that
(3a) is an integer.
(3b) is an integer.
Solutions to above:
Proof (1):
Let , with
and
, and let
,
.
(1a) Proving this inequality amounts to proving that . If
, the result is immediate. If
, then
and therefore
or
and the result follows.
(1b) It suffices to show that . If
, the result follows. On the other hand, if
, then we must show that
. Clearly,
. Now, since
for all
, it follows that
, which gives the result.
(1c) It is enough to show that . If
, the result is immediate. On the other hand, if
, then
and since
, we obtain the result.
Inequalities in (1d) and (1e) are obtained in a similar manner.
Proof (2):
This follows by observing that for each integer
.
Proof (3):
For part (3a), in light of Legendre’s theorem mentioned above, it is enough to show that
, an inequality which is a consequence of Problem (1a) above.
For part (3b), use, again, Legendre’s theorem and result of Problem (1c).
More classic gems of number theory are in the mines!! 🙂
–Nalin Pithwa