**divisibility
**

We say that *divides* , or that is a *multiple* of , if there is an integer such that . In this case, we write and say that is a *divisor* of . If does not divide , we write .

e.g. , so and .

Knuth et al in Concrete Mathematics use and say “The notation `m|n’ is actually much more common than `m\n’ in current mathematics literature. But vertical lines are overused – for absolute values, set delimiters, conditional probabilities, etc. – and backward slashes are underused. Moreover, `m\n’ gives an impression that m is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward.”

**greatest common divisor**

The greatest common divisor of two positive integers and , written , is the largest number that divides evenly (no remainder) into both. If and have no common factor, .

**factorial !**

e.g.

is defined as

**floor/ceiling functions**

means rounded up to the next-highest integer.

means rounded down. e.g. and