adding numbers that are products of primes

Question

The title is rather moot, didn't find better words to describe it, so please bear with me...

Consider two integers

a = p1 * p2 * p3 .... * pn
b = p(n+1) * ... * pm

where p1 till pm are distinct primes. Then none of this primes appears in the factorization of the sum of the two numbers, ie

mod((a + b), pi ) > 0    for all pi = p1....pm

I only stumbled upon this fact by chance and now I wonder: Is there some theorem about this? Does this property have a name?

Maybe this is just too trivial to be a theorem or even get a name, in that case, I'd be happy to get any pointers about where I can read more on it.

Answer 1

If $p_i\mid a+b$ and $p_i\mid a$ then $p_i\mid a+b-a=b$. Contradiction.

Answer 2

Yes it's true. It follows from the opposite of the distributive property of multiplication over subtraction (addition of negatives). That is, if $r\mid q=a+b$ and $r\mid a$ then $r\mid q-a=rc-rd=r(c-d)$ . More generally, if $r\mid q=a+b$ then a and b , are additive inverses (sum to 0) mod r. The property we used, is simply the case of them both having remainder 0.