**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*

### Like this:

Like Loading...

*Related*