 Ramagopta also solved Chinese remainder-type congruences where a number had a given relationship to several divisors. This could be done by solving the congruences pairwise and interpreting the results as a single congruence. For example, let's say we want to find a number that leaves remainder 5 when divided by 12, 7 when divided by 31, and 3 when divided by 11. So we already found that 317 leaves remainder 5 when divided by 12 and 7 when divided by 31. The important insight here is that since the remainder when divided by 12 or 31 remains unchanged if we add 12 times 31, 372, we can solve a new problem, find a number that leaves 317, when divided by 372, which takes care of the first two congruences, and 3 when divided by 11, which incorporates the last congruence. And we can solve this problem exactly as we've solved the others. So again, we'll set up our table with the greater remainder on the right. We'll perform our divisions. Again, since our last summer difference is negative, we should begin with the next to last column to avoid negative clever numbers. And our first clever number can be 0. Our first clever remainder is our clever number times our remainder plus the summer difference, which is in fact divisible by the divisor. Going back a step, the previous clever number, the clever number times the quotient plus the clever remainder. Our previous clever remainder is our new clever number times the remainder plus the summer difference divided by the divisor. The previous clever number, times the quotient plus the clever remainder. And again, we don't need it, but it's a useful check if we can compute it. The previous clever remainder is our new clever number times the remainder plus the summer difference. Then divided by the divisor. And again, the fact that we can get a clever remainder says that we're on the right track. And since this is our last column, we can find our solution directly. Our clever number times our divisor plus the remainder, which clearly leaves a remainder of 317 when divided by 372. And we also note we leave remainder 3 when divided by 11.