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 .

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The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

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