ABC soup

We know that:
• $a,b,c$ are positive numbers.
• $b(a+c)=518$
• $a(b-c)=360$

Now lets factor $518$ and $360$ .
$518=2×7×37$
$360=2×2×2×3×3×5$

Values that can be obatined from $360$ are $2,3,4,5,6,8,9,10,12,15,18,20,24,36,40,45,48,72,90,.....$

Now lets take $a+c=74$ (Multiplying $37×2$ ).
Now $b$ should be $7$ .

Now, since $b$ is $7$ .
$a(7-c)=360$
Now putting any value obtained from $360$ and keeping in mind that $a$ should be any number obtained from $360$ .

We get $c=2$ and $a=72$ .

Therefore $a=72$ , $b=7$ and $c=2$ .

Product of $abc=72×7×2=\boxed{1008}$