 Suppose we're working mod 7. Since 7 is equivalent to 0, we know that 30 is equal to 23 plus 7, but we could ignore the 7. And 23, well that's 16 plus 7, and again, since 7 is equivalent to 0, we could ignore the 7. 16 is 9 plus 7, and ignoring the 7. 9 is 2 plus 7, and ignoring the 7. And so 30 is equivalent to 23, 16, 9, or 2. But wait, there's more. I can rewrite 2 is negative 5 plus 7, and so I have another equivalent, negative 5. And it should be clear we can keep going. 30, which is congruent to negative 5, well negative 5 is congruent to negative 12 plus 7. And so we can generate an infinite number of these congruences. And since they're all equivalent, we can use any one of them. But which one do we want to use? And any sensible person would say, pick the largest and most inconvenient value. Wait, wrong script. Sorry, any sensible person would say, well, if we have a choice, let's pick a positive value that is as small as it could be. So let's pick this value 2 because it's the least positive value. And that leads us to an important problem. Given A, find the least non-negative B for which A is equivalent to B mod N. So let's make a few observations. First thing we note is that our definition of division with remainder, suppose N is a positive integer and A, B, and R are integers where the relationship A equals BN plus R with R between 0 and N. Then A divided by N is equal to B with remainder R. Now remember, if you're working mod N, N itself is equivalent to 0 mod N. And so this means that A, which is equal to BN plus R, is going to be congruent to R itself. And this gives us a useful way of finding a number mod N. It's the remainder when we divide by N. And let's think about this a little bit more. This remainder is always going to be less than the divisor. And consequently, working mod N means never having to work with large numbers. The largest number we'll ever have to work with is less than N. For example, let's say we want to find the least number congruent to 153 mod 23. Well, let's divide. 153 divided by 23 is... And so 153 is going to be congruent to the remainder mod 23. Let's play around with this a little bit. Suppose A is congruent to R mod N, and let B be congruent to R mod N as well. And so let's think about the relationship between A and B. So since A is congruent to R mod N, this means that A divided by N leaves remainder R. And so we can write A as P times N plus some remainder R. Similarly, because B is congruent to R mod N, we know that R is the remainder when B is divided by N. And so we know that B is something times N plus R. Now, I can stare at this for a few minutes and note a couple of things. I don't want to multiply it. I don't want to add it. I don't want to divide. But if I subtract A minus B, I get PN minus QN, and I could factor N times P minus Q. And this leads to a very useful result. Suppose that A is equivalent to B mod N. They have the same remainders, in other words. Then A minus B is a multiple of N. And I should stop here, because we want to think about things in terms of multiplication. Unfortunately, we never do. And so the other way we can look at this is A minus B is divisible by N. And again, you really don't want to think about this divisibility property. You really want to think about this multiplication property that A minus B is something times N. It's a much more useful, much easier way of looking at the relationship. We can also go backwards. Suppose A minus B is divisible by N, then A must be equivalent to B mod N. So for example, let's try to determine which of these are congruent mod 37. So we could try to find the remainders when all of these are divided by 37, but that's a lot of work. So rather than try to find the remainder when these are divided by 37, we consider the difference between the values. For example, 5,072 minus 4,9098 is 74. And since 74 is 37 times 2, then the difference is a multiple of 37. And so the two values are congruent mod 37. Well, that worked pretty well. Suppose I find the difference between two of the other terms. So say 5,072 minus 5,050. And we find the difference is, and since 22 is not a multiple of 37, then 5,072 is not congruent to 5,050 mod 37. And finally, we have one last difference to find. 5,050 minus 4,998, and that works out to be 52. And since 52 is not a multiple of 37, then 5,050 is not congruent to 4,998.