Last time we showed that given (integers) with , there are with such that . We began today by showing that these integers are unique. When , we say that divides , in symbols .
Definition. A greatest common divisor of the integers not both zero, is a positive integer that divides both and such that any integer that divides both , also divides .
We showed that for any not both zero, there is a unique such , in symbols or simply . We also showed the following characterization:
Theorem. Let be integers, not both zero. Let for some integers , . Then the following are equivalent statements about the integer :
- , , and .
- is the least member of .