Given a real number x, the expression represents the largest integer . Also, we denote by the closest integer to x:
Let x and y be real numbers. Then,
(i) , if m is an integer;
(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
(2) Show that is an even integer for each .
(3) Let . Show that
(3a) is an integer.
(3b) is an integer.
Solutions to above:
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.
This follows by observing that for each integer .
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!! 🙂