Sum of factors 2

In the last video, we kind of reasoned our way towards finding the sum of the factors of 27,000. What I want to do in this video is really just revisit what we did in the last video, and see if we can come up with a more generalized way of finding the sum of the factors of a number. So in the last video, we started with 27,000. And we figured out that its prime factorization is 3 to the third, times 2 to the third, times 5 to the third. Or we could have said 2 the third-- we could have switched the orders here. And we used that information to say, well, OK, if this is the prime factorization of 27,000, than any factor-- so, let's say that this is some factor of 27,000-- it has to be the product of powers of 3, 2, and 5. So let me make that clear. It has to be 3 to something times 2 to something else-- they could be the same number-- times 5 to something else. And each of these x, y's, and z's have to be between 0 and 3, and they could be equal to 0 or 3. And actually, make it be clear-- they have to be integers. They have to be integers, where these are integers. Integers. This notation here implies that you could be any number there, but they have to be integers in between 0 and 3, including 0 and 3. And just to be clear, we can just generate arbitrary factors here. We could say, OK, let's pick zero. It could be 3 to the 0, times 2 to the second power, times 5 to the 1. This will be a factor of 27,000. You could figure out this is 1, times 4, times 5. This is equal to 20. And you can just look-- you can just keep taking different values of x, y, and z to get all of the factor. So essentially, you have to through all of the different combinations here and add them up. And that's what we did in the last video. And to be able to think through that, I drew a little bit of a grid. I started with powers of 3. So 3 to the 0-- let me write it here. So you start with 3 to the 0, 3 to the 1, 3 squared, 3 to the third is 27. And then I had powers of 2-- 2 to the 0, 2 to the first, 2 squared, 2 squared is 4. And then 2 the third is 8. And here, I was just looking for the combinations where we're looking at the powers of 3 and 2 and we're multiplying it times 5 to the 0. So, we're not even looking at having a 5 as a factor yet of all of these combinations. And what we did is we just took each of these numbers and we multiplied them by each other. So these would just be a 1, a 3, a 9, and a 27. I just took 1 times 1 is 1, 3 times 1 is 3. And then you could go down this row-- 1 times 2 is 2, 3 times 2 is 6. 9 times 2 is 18. You could just keep going-- 27 times 2 is 54. And you could keep figuring out all of these. But the shortcut that we did in the last video is we said, look, you don't have to figure out all of those. You could just take this sum over here-- so whatever this is. So 1 plus 3, plus 9, plus 27. We figured out in the last video it was 40. This is the sum for this first column. The second sum is just going to be 2 times this number. So let me make it clear. This is the same thing as 1 times this. The second column is going to be 2 times this thing. Because each of these terms are just being multiplied by 2. So it's 2 times 1, plus 3, plus 9, plus 27. And then the third column is going to be 4 times. It's going to be 4 times this business over here. So 4 times-- let me scroll over-- 4 times 1, plus 3, plus 9, plus 27. And then the fourth column will be 8 times, and then they'll be plus 8 times all of that stuff. 1 plus 3, plus 9, plus 27. In the last video, I added them up and got 40, but I'm keeping them this way on purpose. Because this thing right here, what I just wrote-- and this is essentially the sums of all of the combinations where our x's and y's are integers between 0 and 3, including 0 and 3. And z is 0, because we're not multiplying any of these by 5. So when we ignore the 5, the sum here-- and this is exactly what we did in the last video-- this is exactly equal to 1 plus 3, plus 9, plus 27, times 1 plus 2, plus 4, plus 8. If you just take this term right here and distribute it onto each of these terms you will get this expression up here. So to find all of the combinations of the powers of 2 and 3 with not multiplying things by 5, you, literally, just have to take the powers of 3 up to the third power, because that's how many powers-- that's how high we get when we do the prime factorization of 27,000-- and sum them up. And then multiply that times the power of 2 up to the third power. And then, what we did in the last video, we figured out. OK, this value right here-- this is when we're only dealing with 5 to the 0 power. If we want all of the combinations, we would want this grid multiplied by 5 to the 0, which is essentially what this grid already is. And then we would want to add that to that grid multiplied by 5 to the first. And then, add that to this grid multiplied by 5 squared. And then add that to this whole grid multiplied by 5 to the third. And so this whole grid-- the sum of this whole grid-- is this right over here. Let's just call this, I don't know. For simplicity, let's just call this A. So if you really want the sum of all of the factors of 27,000, you would take A and multiply it by 5 to the 0, which is 1. And then add that to A multiplied by 5 squared-- or sorry, multiplied by 5 to the first, which is 5. And then add that to A multiplied by 5 squared, which is 25. And then add that to A multiplied by 5 to the third, which is 125. And of course, you can factor out an A here. So this is our total sum. This would be the sum of all the factors of 27,000. But you could factor and A out here. And so our sum would be A times 1, plus 5, plus 25, plus 125. And we know what A is. A is this thing over here. So the sum of all of the factors of 27,000-- actually, let me keep this over here-- is equal to-- and I'll color code it-- it's equal to-- this is A. So A is 1 plus 3, plus 9, plus 27-- it's not the right shade of green, but hopefully, you get the point-- times 1 plus 2, plus 4, plus 8, times 1 plus 5, plus 25, plus 125. So this, essentially, is giving you a formula for finding the sum of the factors of a number. You do the prime factorization of the number. And I'm not doing in general terms, I'm doing with a particular case. And I'm doing that because I think it makes it a little bit clearer why this works. But our prime factorization of 27,000 is 3 to the third, times 2 to the third, times 5 to the third. If we really want to do the sum-- and this is exactly what we did in the last video, I'm just expressing it slightly different here-- we just take all of the powers of 3 up to 3 to the third, add them up. All the powers of 2 up to 2 to the third, add them up. All the powers of 5 up to 5 to the third, add them up. And then, take the product. And we can do that right now. So this is going to be-- so this first term over here is 40, this term right here, 40 times-- this is 4 plus 8 is 12, plus 3 is 15. Right? 3, yup. 15. And then times-- 25 plus 125 is 150. So this is 156. And if you take the product-- here, actually, let's just do it just to prove the point that this actually does work. So 40 times 15-- 4 times 15 is 60-- so this is going to be 600. So this part right here is going to be equal to 600. So it's going to boil down to 600 times 156. So let's just do that. 156-- let's just multiply it by 6, and then I'll add two 0's to it at the end. So 6 times 6 is 36. 6 times 5 is 30, plus 3 is 33. 6 times 1 is 6, plus 3 is 9. And of course, if we want to multiply it by 600, we just add two 0's here, and we get the exact same answer that we got in the last video. So hopefully, you found that useful. I haven't proven it in general terms, but you can-- maybe I'll do that, if I feel like doing a video on that. But you see a general method for finding the sum of the factors of a number. Actually, let me just write that. Let's just do another problem just real fast. Let's say I had the number-- I don't know-- let's say we had to find the sum of the factors of 300. So 300 is 3 times 100, which is 10 times 10. And that is 2 times 5, times 2, times 5. So 300 is equal to 3 times 2 squared, times 5 squared. That's the prime factorization of 300. So if we wanted to find the sum of all of its factors, it would be 3 to the 0, which is 1, plus 3 to the 1, which is three. We won't go to any higher powers of 3 here, because that's the highest power we have in the prime factorization. So that's dealing with the 3's. Times 2 to the 0, which is 1, plus 2 to the 1, which is 2, plus 2 squared-- that's how high we can get, which is 4. And then, times-- you just do the 5-- times 5 to the 0, which is 1, plus 5 to the first power, which is 5, plus 5 squared, which is 25. Can't go any higher than that, because we only go up to two 5's up here at 5 to the second power. And so this is going to be 4 times 7, times 31. So this is going to be 28 times 31. 1 times 28 is 28. Put a 0 down here. 3 times 8 is 24. 3 times 2 is 6, plus 2 is 8, and then you get 8, 6, and 8. So the sum of all the factors of 300 is 868. Hopefully, you found that interesting.