If 24 is a factor of H and 28 is a factor of K

[griffin.811]

My question stems from the chapter 1 problem set, question 8, in the 2005 number properties book.

The question asks "If 24 is a factor of H and 28 is a factor of K, must 21 be a factor of HK?" I have no issues with this.

However, if the question asked "...must 21 be a factor of H+K?" would the H+K prime box still include 2,2,2,2,2,3, and &7?

If not, what would the H+K prime box consist of?

Thank you

[RonPurewal]

If you add or subtract numbers, then any primes common to those two numbers will also be factors of the sum/difference.
E.g., if you add or subtract two multiples of 5, you'll get another multiple of 5.
As for the rest of the primes, there's no definite relationship.

So, in this case, the only primes common to 24 and 28 are 2 x 2. (You can also see this without having to do the prime factorizations; the largest number that goes into both of them evenly is 4.)
So, if you add a multiple of 24 and a multiple of 28, or if you subtract two such numbers, the best you can do (in general) is to say that you'll get a multiple of 4.

[RonPurewal]

If the two numbers don't have any common factors, then anything whatsoever can happen.
For instance, if you add 32 (= 2 x 2 x 2 x 2 x 2) and 15 (= 3 x 5), which have no common factors other than 1, the result is 47. 47 is a prime number, so it's clear that none of these prime factors are retained.