A little terminology:

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.

These distinctions are necessarily subjective.

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

\mathbb\[Z\]=\lbrace \ldots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \ldots \rbrace

The 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.

Not all number are integers. If we need to round such numbers to integers, we use 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

(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 .

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.