 All right so let's take a look at problem 319. So this asks a question which of the numbers here, here, or here are divisible by seven and since we know the definitions we can do mathematics. Oh wait remember don't watch this video. Remember a main goal of our course is to develop your problem-solving ability and you only ever have one opportunity to solve a problem. Now some of you might wonder well you know why aren't I just teaching you how to answer these questions and the reality is if I show you how to solve a problem it's because I don't believe you are smart enough to figure it out on your own. You should really think about what that means. It means if I say here are the steps that you need to answer this question. What I'm also saying is I don't think you're smart enough to figure this out. On the other hand if I give you a problem and ask you to solve it without telling you how to answer the question step by step what it means is I think you can figure this out. I think you are smart enough to figure these problems out on your own. So ideally these videos no one will ever watch them. Nobody watches them because they can figure them out and my assumption that you can figure these questions out that you can solve these problems is the correct one. So hopefully no one is watching these videos. So again hopefully no one is actually watching this video and I'm talking to the ether but let's see if we can answer this question. Well we know the definitions so we can do mathematics. If you don't know the definitions you cannot do mathematics you can push symbols around on paper you can get the right answer but you will never be doing mathematics. You have to know the definitions. I can't emphasize that enough. Now in this case the question involves are these which of these numbers are divisible by seven. So we go back to what it means to be divisible by a number and by the definition a number is divisible by seven if we can write it as a product of seven and whatever else there is. So let's take a look at this. So here q our number this is this horrible messy thing and the thing to notice here is that there's this last term here that tells us we have 25 factors of seven. So if you want to think about this this q has a whole bunch of numbers multiplied together and the tail end of that product includes 25 factors of seven. So certainly associativity and commutativity of multiplication being true I can think about q as being a product of seven and a whole bunch of other things. So certainly q is going to be divisible by seven. What about the others? Well the fundamental theorem of arithmetic guarantees that the prime factorization of a number is unique up to the order of the factors. So seven is prime and so I know that if the prime factorization of a number does not have a seven then that prime then the factorization of that number can never have seven as a factor. If I don't have a seven I can't get it in. Well so that means I need to find the prime factorization of each number and see which ones have sevens and which ones don't. So let's take a look at n. So first off n, 4 to the 30th, 25 to the 70th. There's no 7s there but 4 and 25 aren't prime. So the fact that there are no 7s there doesn't mean anything because this is not the prime factorization. But I can find the prime factorization that's 2 to the 60th, 5 to the 140th and now 2 and 5 are primes. There's no 7 here so I know that 7 is not divisible. 7 does not divide n. Well what about m? Again no 7s here. Well this is not a 7 this is actually a 70 but again 30 and 70 aren't prime numbers so I can find the prime factorization and well I have a bunch of 7s here. This is 7 to the power 25. There's 25 factors of 7 so there's lots of 7 so 7 does divide m and you should check the others on your own.