Suppose a, b, and c ∈ Z and that gcd(a, b) = 1.
If a | bc, then what is the gcd(a, c) and the gcd (b, c) ?

Question

Suppose a, b, and c ∈ Z and that gcd(a, b) = 1. 
If a | bc, then what is the gcd(a, c) and the gcd (b, c)  ?

rational · Answer

a | bc   and gcd(a, b) = 1

=> a | c
=> gcd(a, c) = a

rational · Answer

for second part :

a|c  =>  c = ak

gcd(b, c) =  gcd(b, ak) = gcd(b, k) = gcd(b, c/a)

anonymous · Answer

a|c means a is a divisor of c ... correct? and a | bc means a is a divisor/factor of b * c? not even sure of notation? any suggestions of resources to better understand this notation?

rational · Answer

you're right ! you can also "read" it as :

a | b  

means

a divides b

rational · Answer

where ever you see "|", replace it with the word "divides"

anonymous · Answer

a divides b ? gawd...why has it taken me so long to see that notation?

rational · Answer

yes to be more accurate :

a|b  means :   "a divides b" evenly without any remainder...

anonymous · Answer

does it mean "divides evenly" you read my mind:)

rational · Answer

or "a goes into b" evenly without any remainder

anonymous · Answer

how do I give you a medal:)

rational · Answer

yep ! but i read it as "a divides b"  ... evenly is implicit :)

anonymous · Answer

thank you so much!

rational · Answer

download this free pdf : http://worldtracker.org/media/library/Science/680%20maths%20books/Number%20theory/Elementary%20Number%20Theory%20-%20David%20M.%20Burton.pdf

rational · Answer

refer to 2nd chapter

anonymous · Answer

just started Abstract Algebra...I have heard it is a very hard course...even harder than linear alg or calc 2

rational · Answer

oh i never took abstract algebra before.... im into engineering... but i dont think its tougher than analysis lol..

anonymous · Answer

some day...i will take analysis:)

rational · Answer

for me number theory itself is hard lol, let alone even think of abstract/analysis -.-

rational · Answer

most ppl who take NT, abstract and analysis claim NT is the easy one :/

anonymous · Answer

which of these must be true of my first question?  it has to be b right?
a. gcd(a, c) = 1. 
b. gcd(a, c) ≠ 1. 
c. gcd (b, c) = 1. 
d. gcd(b, c) ≠ 1.

rational · Answer

Yep !

rational · Answer

by any chance is it mentioned that $a \ne 1$ for this question anywhere  ?

anonymous · Answer

thanks again man:)

anonymous · Answer

no...it is not mention anywhere...that $$a \neq 1$$

anonymous · Answer

I think that is why they put MUST in the question...

anonymous · Answer

@rational did you ask me for help with something...I was at lunch