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calculate the sum of all positive divisors of 27,000 the easiest thing that I can think of doing is first take the prime factorization of 27,000 and then that'll help us kind of structure our thought of what all of the different divisors of 27,000 would have to look like so 27,000 27,000 is the same thing as 27 times 1,000 which is the same thing as 3 to the third times 10 to the third and 10 is of course the same thing as 2 times 5 so this is the same thing as 2 times 5 to the third or so same thing as 2 to the third times 5 to the 3rd so 27,000 27,000 is equal to 2 to the third times 3 to the third times 5 to the third so any divisor of 27,000 is going to have to be made up of the product of up to three twos up to three threes and up to 3/5 so let's try to look at all the combinations and think of a fast way of summing them so let's just say it has no fives in it it has no fives in the divisor so if it has no fives then it can have up to three up to three twos so let's say it has zero twos so I'm just going to take the powers of two so if it has zero to s then we'll put a 1 here if it has two twos and it has to be just little by 4 if it has three twos it's going to be divisible by 8 when I say three twos I mean two times two times two now let's do it with the threes let's do it with the threes if you have oh wait I forgot a power if you have 0 2 so it means it's just divisible by 1 from looking at the twos if you have two - if you have one two it's divisible by just 2 if you have two twos you're divisible by 4 and if you have three twos and what I mean that I'm saying 2 times 2 times 2 that means you're divisible by 8 now let's do the same thing with 3 from the point of view of 3 if you have no 3s that means at least you're divisible by 1 if you have 1 3 that means you're divisible by 3 2 threes or 3 times a 3 means you're divisible by 9 if you have three threes it means you're divisible by 27 so let's look at all of the possible combinations and for this grid that I'm going to generate right here we assume that you're not divisible by five or you're only divisible by five to the zero power so what are all the possible numbers here well you have one times one is one that's divisible by one and one you have 1 times 3 which is 3 1 times 9 which is 9 1 times 27 which is 27 so these are all the numbers that are divisible by that have up to three threes in them from zero to three threes in them and they have no twos in them if you throw another 2 in here you're essentially going to multiply all of these numbers by two if you throw another two in here going to multiply all of these numbers by two now before I do this because I want to do this as fast as possible I could figure out what these numbers are I could multiply them but instead let's just take the sum let's just take the sum here of this row of this first row that we just did we have 1 plus 3 plus 9 plus 27 3 plus 27 is 31 plus 9 is 10 so this is going to be 40 now whatever these numbers are they're all going to be two times these numbers so the sum is going to be 80 and the sum over here is going to be 2 times the previous row because here we multiplied by 2 here were multiplying by 4 so it's going to be 160 and over here we just multiply by 2 again it's going to be 320 or another way of thinking about it whatever the sum is here it's going to be 8 times the sum of the first row and I could just so you know what I'm doing I could actually put numbers here this number here would be 8 24 72 and whatever 8 times 27 have with that 161 60 Plus 56 so it's 216 but we don't want to do that we just have to think about the sums so if you think about all of the divisors of 27,000 that are not divisible by 5 so they're only divisible by 5 to the 0th power I guess you could say it we have now figured out there some it's going to be the sum of all of these rows so if you take 40 plus 80 you have 120 plus 160 is 280 plus 320 is is 600 so this is the situation these are this is the sum of all of the combinations of the two that don't have any fives in them now if you took the same combination of twos and threes if you took the same combination of twos in so these added up to 600 let me write it over here these added up to 600 so no fives no fives now if you did the same exact thing that we just did here but we just multiplied everything by five so we then be looking at all of the combinations that have this many twos in this many threes and one five what would happen to this sum well we would just we would just multiply it by five so let's multiply that by five so you multiply 600 by five you get 30 with two zeroes and so this is one five one five in the prime factorization of the divisors one one five just like that now if I wanted two fives I can just multiply by five again so if I multiply by five again I get fifteen fifteen thousand fifteen thousand this is two fives - five another way of thinking about this if I just multiply this whole every term here by 25 we're essentially multiplying by five times five this sum is going to be 600 times 25 which is fifteen thousand now if I have three fives if I have three fives then I can just multiply this by five again five times 15 is 50 plus 25 is 75 so it's 75 75 thousand so now I know all the sums if I have no fives the sum of all the divisor is six hundred I have one five three thousand so on and so forth so if I want the sum of everything I just take the sum of these numbers let me scroll down a little bit so I get well I have 0 0 then in the hundreds place I only have a six and then three plus five plus five is thirteen is that right yep that's 13 and then carry the 1 and then I have a 9 so I have 93 thousand and six hundred so the sum of all positive divisors of 27,000 93 thousand six hundred hope you found that entertaining