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Current time:0:00Total duration:10:54

Video transcript

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 the last video we started with 27,000 we figured out that its prime factorization is 3 to the third times 2 to the third times 5 to the third or because the 2 to switch the order is here and we use that information say well ok if this is the prime factorization of 27,000 then any factor then 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 2 something times 2 2 something else they could be the same number times 5 2 something else and each of these X wise and Z's have to be between 0 and 3 and they could be equal to 0 or 3 an axiom it can be clear they have to be integers they have to be integers where these are integers integers this notation kind of 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 okay let's pick 0 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 could just look you can just keep taking different values of x y&z to get all the factors so essentially you have to go 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 to the third is eight and here I was just looking for I was just looking for the combinations where we're looking at the powers of three and two and we're multiplying it times five to the zero 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 we took so these would just be a 1 a 3 a 9 and a 27 I just took 1 times 1 isn't 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 but what that 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 is going to be 2 times 2 times this thing right because each of 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 4th column will be 8 times and then there'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 this is essentially the sums of all of the combinations where X is where our X or our X's and Y's are between 0 & 3 integers between 0 & 3 including zeros in 3 and Z is 0 because we're not multiplying any of these by 5 so when you we ignore the 5 the sum here and this is exactly what we did in the last video this is exactly equal to this is exactly equal to one plus three plus nine plus 27 times x one plus two plus four plus eight if you just take this term right here if you just take this term right here and distribute it onto each of these terms and distribute it onto each of those terms you will get this expression up here so to find all of the combinations of the powers of two and three with the with not multiplying things by five you literally just have to take the powers of three up to the third power because that's how how many powers that's how high we get when we do the prime factorization of twenty-seven thousand and sum them up and then multiply that times the power of two up to the third power and then what we did in the last video we figured out okay this value right here this is when we this is when we're only dealing it with five to the zero power if we want all of the combinations we would want this grid multiplied by five to the zero which is essentially what this grid already is and we would want to add that to that grid multiplied by five to the first and then add that to this grid multiplied by five squared and then multiplied and then and then add that to this whole grid multiplied by five 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 this 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 1st which is 5 and then add that to a multiplied by 5 squared which is 25 and then add that and then add that to a multiplied by 5 to the 3rd 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 of the factors of 27,000 you could factor an 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 this is a so a is 1 plus 3 plus 9 plus 27 it's not the right shade of green but I hopefully you get the point times 1 plus 2 plus 4 plus 8 times 1 plus 5 plus 25 plus 1 plus 125 so this really just kind of sit this is 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 it in general terms I'm doing it 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 3rd add them up all the powers of 2 up to 2 to the 3rd add them up all the powers of 5 up to 5 to the 3rd 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 this is 40 this term right here 40 times this is 4 plus 8 is 12 plus 3 is 15 all right 3 up 15 and then times times 25 plus 125 is 150 so is 156 156 and if you take the product here actually let's just do it just to 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 well it's just multiplied it by 6 and then I'll add two zeros 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 zeros here and we get the exact same answer that we got in the last video so hopefully you found that useful I'm not I haven't proven it in general terms but you can't maybe I'll do that I feel like doing a video on that but you see a general a general method for finding the sum of the factors of a number actually let me just let me just write that let's just do another problem just to 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 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 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 3 to the 0 which is 1 plus 3 to the 1 which is 3 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 threes times 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 and then times we 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 fives up here 5 to the second power and so this is going to be 4 times 7 4 times 7 times times 31 so this is going to be 28 times 31 28 times 31 1 times 28 is 28 but a 0 down here 3 times 8 is 24 3 times 2 is 6 plus 2 is 8 and then you get 8 6 & 8 so the sum of all of the factors of 300 is 868 hopefully you found that interesting