Monk of Charles the Great

Charles the Great truly earned his title, as he cared about the future of his country's people. In one of the first known schools in Aachen, he invited the monk Alcuin to teach mathematics in the year $795$. Alcuin proposed the following problem to his students:

"$100$ sheffels (units of currency) were divided among men, women, and children. Each man received $3$ sheffels, each woman received $2$ sheffels, and each child received half a sheffel. How many men, women, and children were there?"

At that time, the general solution of linear Diophantine equations was not known, and only a few specific solutions were sought that satisfied the conditions. Alcuin himself provided only one solution: there were $11$ men, $15$ women, and $74$ children. In reality, the problem has $784$ solutions in natural numbers.

We suggest that you continue to study Diophantine equations and solve a similar problem in the following form:

"$C$ sheffels were divided among men, women, and children. Each man received $A$ sheffels, each woman received $B$ sheffels, and each child, just as in ancient times, received half a sheffel."

How many solutions in positive integers does this problem have?