Thread: greatest common divisor

Can anyone please help me with this proof?

" If a|bc, then shows that a|[gcd⁡(a,b)∙gcd⁡(a,c)] "

Any suggestions will be very appreciate.

Originally Posted by deniselim17

Can anyone please help me with this proof?

" If a|bc, then shows that a|[gcd⁡(a,b)∙gcd⁡(a,c)] "

Any suggestions will be very appreciate.

Since the gcd of any two numbers can be written as a linear combination we can rewrite each of the above as follows

and



also since

so

Grouping we get...(and subbing in from above)



so finally