Determine if it is multiple I'm solving an exercise that says, of 2 numbers

Question

I'm solving an exercise that says:

It can be determined that (A + B) is a multiple of 7, if it is known that:

(1) A is a multiple of 4 and B is a multiple of 3

(2) The difference between A and B is a multiple of 7.

I know that,

An number A is multiple of another number B, when:

A = kB, where k is an integer.

Then I solved it:

Image of my development

I thought that was fine, but the answer is "Additional information is required", then could you tell me what, I was wrong?

Answer 1

From the fact that $A$ is a multiple of $4$ and that $B$ is a multiple of $3$, you deduce that there are natural numbers $k$ and $k'$ such that $A=4k$ and that $B=3k'$. Your error was that you assumed that $k=k'$.

Take, for instance, the case in which $A=24$ and $=3$. Then $A$ is a multiple of $4$, $B$ is a multiple of $3$, $A-B$ is a multiple of $7$, but $A+B$ isn't a multiple of $7$.

Answer 2

$A$ is indeed $4$ multiplied by some integer, and $B$ is $3$ multiplied by some integer, but it doesn't have to be the same integer. So $A+B$ isn't $4K+3K$, but rather $4K+3L$.

Answer 3

First consider B being a multiple of 7. If so, then for A+B to be a multiple of 7, A must also be a multiple of 7. Therefore one set of valid pairs for (A, B) is (28a, 21b); a,b are integers.

Next consider B, not a multiple of 7. If so, then for A+B to be a multiple of 7, A cannot be a multiple of 7 either. Now, if either A+B or A-B is a multiple of 7, the other cannot be because the difference between them is 2B, which cannot be a multiple of 7.

Only valid pairs are therefore (28a, 21b); a,b are integers. So, you also need to know that either A or B is a multiple of 7.