Question

Eleanor claims that the GCD(0,a)=0. Is she correct? What does she understand about GCD? What does she not understand?

Answer 1

SHE is correct for GCD(0,a)=0 if a=0, becuase the gcd does not refers to largest in the ordering of the natural numbers, but to being largest in the partial order divisibility on the natural numbers.

where we consider a to be largest than b only when b divides evenly into a.Most of the times these 2 orderings agree whenever the second is defined. However 0 is the smallest natural number, under the divisibility order 0 is the greatest natural number because every number divides 0.

therefore every natural number is a common divisor of 0 and 0. then 0 is the greatest of the natural number in divisibility so gcd(0,0)=0.

similarly gcd(0,n)=n for all n N including n=0.

in our situation gcd(0,a)=a it should not be gcd(0,a)=0.

so she is incorrect.