15. Find all positive integer solutions \(a, b, c\) that satisfy \((a, b, c)=10,[a, b, c]=100\).
15. $(a / 10, b / 10, c / 10)=1,[a / 10, b / 10, c / 10]=10 . a / 10, b / 10, c / 10$ can only take the values $1,2,5,10$ and satisfy the above two conditions. There are three possible scenarios: (i) $a / 10=b / 10=c / 10$, which is impossible; (ii) Two of the three are equal, and the other is uniquely determined by the conditions, they are $\{10,10,1\}$, $\{5,5,2\},\{2,2,5\},\{1,1,10\}$, and permutations, giving a total of $3 \cdot 4=12$ solutions; (iii) All three are distinct, any three distinct values can be chosen, they are $\{10,5,2\},\{10,5,1\},\{10,2,1\},\{5,2,1\}$ and permutations, giving a total of $6 \cdot 4=24$ solutions. Multiplying each number by 10 gives the solutions to the original problem, with a total of 36 solutions.