 One of the more useful extensions of the number concept is modulo n arithmetic, and this works as follows. Suppose I have some positive integer n, greater than 1, and suppose that n divides the difference p minus q of two other integers, which could be positive or negative. Then we have the following. First off, we say that p is congruent to q modulo n. We also would write this using this congruent symbol here. That's three bars looking a little bit like n equals with emphasis. And we would write p congruent to q mod n, and one important bit about the notation. This mod n phrase applies to the entire statement p congruent to q. It does not apply to only the q, it applies to both of them. Or really it's more like a footnote to the entire expression. So if the modulus n is understood, if we know which modulus we're working with, then we will generally just drop it and simply write p congruent to q without writing mod n explicitly. Now, here's a couple of basic properties of modulo arithmetic. These are what you would call fairly obvious properties. You should be able to prove them without too much difficulty. So if p is congruent to q, then q is congruent to p. So remember what that means is that n divides p minus q, so n also divides q minus p. Likewise, if p is congruent to q, then minus p is congruent to minus q. And if p is congruent to q, then k times p is also congruent to k times q. If p is congruent to q mod n and r is congruent to s mod n, again note that we're using the same modulus, then I can take p and add or subtract r, q, add or subtract s, and I still get a congruence. Now in addition to these basic properties of things that are definitely true, there's a number of things that seem like they should be true, but it turns out that these things are actually false in some cases. And one of the most important distinctions between modulo arithmetic and ordinary arithmetic is if I have kp congruent to kq in ordinary arithmetic, I can just drop that factor of k and have p congruent to q. So does it work in modulo arithmetic? Can I drop the factor? Can I drop common factors? Turns out I can't. So for example, 6 times 6, that's 36, is congruent to 6 times 424 mod 12, but I can't drop that common factor of 6, because if I do, I claim falsely 6 is congruent to 4, and that's simply not true. Likewise, another property that's very different from our experience with ordinary arithmetic, if a product is congruent to 0, ordinary arithmetic says that one of those two terms has to be 0. Well, this is also false in modular arithmetic. So for example, 6 times 15 is congruent to 0 mod 30, but neither 6 nor 15 are congruent to 0. So let's take a quick look at this. So prove or disprove 197 congruent to 153 mod 23. And so remember that if a is congruent to b mod n, then n has to divide the difference a minus b. And so, well, the easiest way of doing that, well, let's take a look and see what a minus b actually is. So 197 minus 153 is 44, and we note that 23 does not divide 44, so the congruence is not, in fact, true. Now here's another example. Find the least positive integer a that satisfies 175 congruent to a mod 37. Now since we want a to be congruent to 175 mod 37, then we know that 37 will divide 175 minus a. Let's go ahead and set that up. 175 minus a is 37 times something, and I can rearrange that 175, 37k plus a. And what that means, if I'm looking for the least positive integer that satisfies this equation, then a is going to be the remainder when 175 is divided by 37. So I can find that 175 divided by 37 is 4 with remainder 27, so that tells me that 175 is congruent to 27 mod 37.