(Counting Multiples.)
Let n
and m
be positive integers. Then the number of multiples of m
between 1
and n
is \lfloor \frac nm \rfloor
.

- A
**theorem**is an important result. - A
**proposition**is less important than a theorem. - A
**lemma**is a small result, generally proved before its use in a theorem. - A
**corollary**is a small result, generally a consequence of a recently proved theorem.

Number theory is the study of integers (the 'whole' numbers):

\mathbb\[Z\]=\lbrace \ldots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \ldots \rbraceThe natural numbers are the non-negative integers:

\mathbb\[ N \] = \lbrace 0, 1, 2, 3, 4, \ldots \rbrace
Many number theorists omit 0
from the natural numbers. We do not.

**floor**or

**ceiling**functions. The floor of a number is the integer less than or equal to the number: the floor of 3.142 is 3 . The ceiling of a number is the nearest integer greater than or equal to the number: the ceiling of 1.414 is 2 . We write \lfloor 3.142 \rfloor = 3, \quad \lceil 1.414 \rceil = 2 \\~\\ \lfloor 42.0 \rfloor = 42, \quad \lceil 42.0 \rceil = 42

The number of multiples of m=3
between 1
and 17
is \lfloor \frac\[17\]\[3\] \rfloor = 5
. (\text\[They are \]3, 6, 9, 12 \text\[ and \] 15.)

The number of multiples of m=4
between 1
and 24
is \lfloor \frac\[24\]\[4\] \rfloor = 6
. (\text\[They are \]4, 8, 12, 16, 20 \text\[ and \] 24.)

When we use the term 'between,' we mean it inclusively. The numbers between 21
and 23
are 21, 22 \text\[ and \] 23
.

The number of multiples of m=7
between 1
and 1024
is \lfloor \frac\[1024\]\[7\] \rfloor = 146
.

(Counting Squares.)
Let n
be a positive integer. The number of squares between 1
and n
is \lfloor\sqrt\[n\]\rfloor
.

The number of squares between 1
and 39
is \lfloor \sqrt\[39\]\rfloor=6
. (\text\[They are \]1, 4, 9, 16, 25 \text\[ and \] 36.)

Find the number of squares between 100
and 200
.

The number of squares between 100
and 200
is the number of squares between 1
and 200

**minus**the number of squares between 1 and 99. (Note that if we subtract the number of squares between 1 and 100 , where 100 is a square, we are removing one of the squares that we should be counting between 100 and 200 .) \lfloor \sqrt\[200\]\rfloor - \lfloor \sqrt\[99\]\rfloor = 14-9=5 The squares are 100, 121, 144, 156 \text\[ and \] 169 . (Counting Digits.)
If n
is a positive integer, then the number of digits in the decimal representation of n
is \[ \lfloor \log_\[10\](n) \rfloor + 1\]

(Counting Bits.)
If n
is a positive integer, then the number of digits in the binary representation of n
is \[ \lfloor \log_\[2\](n) \rfloor + 1\]

No proof provided.
(Division with Remainder.)
Let a and b be integers, with b positive. Then there exist integers q and r satisfying
a=q\cdot b+r \textit\[ and \] 0\le r \lt b

q
is known as the

**quotient**and r the**remainder**.