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).