 The Chinese remainder problem requires us to solve problems of the form, find a multiple of n that leaves a remainder of 1 when divided by p. We've been solving this by checking out multiples of n, but can we solve these problems directly? And in fact we can. Chen Zhixiao gives the following procedure, which we'll illustrate with the following example. Suppose we want to find a multiple of 5, that is 1 more than a multiple of 13. To do this, we'll set up a table. So our table will have two columns. The right side will be our multiples 5 and 13, and the left side will be the numbers 1 and 0. And we'll proceed as follows. We'll work the right column, and we'll subtract the smaller number from the larger number as many times as we can, keeping track of the number of times you've subtracted. So 13 minus 5 is 8, that's one time. 8 minus 5 is 3, that's a second time. And we can't subtract any more. So now we'll change our table. We'll multiply these two opposite quarter numbers 1 and 2, and add the result to the third quarter number 0. So 1 times 2 is 2, plus 0 is 2, and our table now becomes this. And lather, rinse, repeat. We'll subtract the smaller number from the larger number and keep track of how many times we can do that. So 5 minus 3 is 2, that's one subtraction. We can't subtract 3 from 2. The product of our quarter numbers 2 times 1 will be added to our third quarter 1, so that's 2 times 1 is 2, plus the 1 that's already there makes 3. And that's our second table. And again, smaller number from larger number and keep track. So 3 minus 2 is 1, one time. Multiply the corners and add smaller number from larger number. Now our end goal is to have the second column be all ones. And so we arrive at that end goal, multiply the quarters. And the way we can read this is that 8 fives, that's 40, is one more than a multiple of 13. So why does this work? Well, let's consider our tables. In our first table, we can read the two lines as follows. The first line says that 1 is 5 more than a multiple of 13. Namely, it's 5 more than 0 times 13. Our second line can be read as 0 fives is 13 less than a multiple of 13. Namely, it's 13 less than 1 times 13. And what this means is that for every 5 we add, we reduce a deficit by 5. And that's because our fives are 5 more than a multiple of 13. So whatever our deficit was, it's now 5 less. And so if we add two fives, we reduce our deficit by 10 and our deficit is now 3. And that gives us the bottom line of the table. 2 fives is 3 less than a multiple of 13. And what that means is that if we add 2 fives, we'll reduce an excess by 3. So 1 fives gives us an excess of 5. 3 fives will give us an excess of 2. And so our top row now indicates that 3 fives is 2 more than a multiple of 13. So if we add 3 fives, we'll reduce our deficit by 2. Our bottom row now tells us that 5 fives is 1 less than a multiple of 13. So if we add 5 fives, we'll reduce the excess by 1. And that gets us to this final table. 8 fives is 1 more than a multiple of 13. And 5 fives is 1 less than a multiple of 13. So let's look for a multiple of 8 that's 1 more than a multiple of 21. So we begin by noting that 1 8 is 8 more than a multiple of 21, namely 21 times 0. And 0 8 is 21 less than a multiple of 21, namely 21 times 1. And so we'll set up our table showing this excess and deficit. Now since 21 is greater than 8, we'll repeatedly subtract 8 and keep track of how many times we've done that subtraction. So we'll subtract once, twice, three times. Oh wait, we can't do that. Now we'll multiply our quarter numbers 1 times 2 and add our other quarter numbers 0. And this result will replace the quarter number 0. And this tells us that 2 8 is 5 less than a multiple of 21. And now we can continue the process. We'll subtract 5 from 8 and we can't subtract another 5. So multiply our quarter numbers and add and replace. And again we can read this 3 8 is 3 more than a multiple of 21. And we can continue. Subtract 3 from 5, multiply the corners and add, replace. And so we can read our new bottom line. 5 8 is 2 less than a multiple of 21. Now subtract 2 from 3, multiply the corners and add, and that tells us that 8 8, 64, is 1 more than a multiple of 21. Now while we have answered the actual question, let's go ahead and finish out the process. So 2 minus 1, multiply and add, and that tells us that 13 8, 104, is 1 less than a multiple of 21.