Hyperbolic geometries arise when this man's fifth postulate does not hold.
One theorem named for this man says that if a prime p divides "a times b," then either p divides a or p divides b. The solutions predicted by the Chinese Remainder Theorem can be found using the extended form of this man's namesake algorithm, which finds the greatest common divisor of two numbers. Hyperbolic geometries arise when this man's fifth postulate does not hold. For 10 points, name this Greek mathematician whose treatise on geometry is the Elements.
One theorem named for this man says that if a prime p divides "a times b," then either p divides a or p divides b.
Euclid