MATH 205 Course Notes- Section 4.1: Even and Odd Numbers
Terminology:
If a × k = b for some integer k, we say that a divides b, and write a|b.
If a|b, then a is a factor of b.
If a|b, then b is a multiple of a.
If a|b, then b is divisible by a.
If a|b, then a is a divisor of b.
Ex. 1: What is the definition of an even number? How many ways can we write a definition for
this?
We can define an odd number in a similar algebraic way, as being of the form 2k + 1 where k is
any integer (could be positive or negative, or zero)
Ex. 2: What do you notice about adding two odd numbers? two even numbers? an odd number
and an even number?
Divisibility
Ex. 3: We know that 4|48 and 4|64. Does it necessarily follow that 4 divides their sum - that is,
does 4 divide (48 + 64)?
In fact, can we say this in general? If a|b and a|c, does a|(b + c)?
It is true! If a|b and a|c, then we can write ak = b and am = c for some integers k and m. Then,
b + c = ak + am = a(k + m), which is a times some integer and so a|(b + c).