Factors and multiples: days of the week

let's say that today and we'll call today day one day one is a Monday what I want to figure out is what is day 300 going to be day 300 what day of the week will day 300 be and I encourage you to pause this video and think about that a little bit so let's just write out the days of the week you have Monday this in a different color you have Monday Tuesday Wednesday Thursday Friday Saturday and Sunday so if we had a lower number here we could just fill out this fill up fill this out Monday is day one Tuesday's day two Wednesday day three four five six seven I'll keep going day eight well that's going to be a Monday again nine ten I'm almost writing a calendar out eleven twelve thirteen fourteen fifteen sixteen so this is kind of useful I could just write it out if I wanted to figure out something you know day 16 or day 20 I could just write that out but this isn't that helpful if I wanted to figure out day 300 or especially wouldn't be helpful if I wanted to figure out day 3000 so can I come up with some mathematical way of thinking about what day 300 is going to be well when I as you see when I started drawing this grid here this grid has rows and in each row you have seven days in it and that makes sense there are seven days of the week so is there a way that if someone just gave you 16 without drawing this grid that you would know that 16 is a Tuesday well one way to think about it that might jump out at you is you could divide 16 by 7 that will tell you how many of these rows will come before 16 that will tell you how many rows come before 16 so 16 divided by 7 is 2 you have 2 rows before 16 right over here you could get 7 into 16 2 times and then you have a remainder and then you have a remainder what is the remainder when you divide 16 by 7 16 divided by 7 well is going to be 2 two times seven is fourteen you're going to have a remainder of two so when we divide we we've historically cared more about this two we normally care more about well how many times does it go into it but now the remainder is actually interesting the remainder is really interesting here because the remainder tells you the the first two just tells you well seven goes into 16 two times that's how many rows you have before getting to the sixteen but then the remainder tells you well in that row where is the 16 so the 16 is remainder 2 so 16 is not the first it's the second entry in the third row and so it's going to be a Tuesday Tuesday is the second day I know what you're saying does that always work well let's try it out with some some other examples let's imagine day 25 day 25 so let's just divide 25 by 7 so I'll do it right over I could do it I make sure I have enough space so if I have 7 goes into 25 it goes 3 times 3 times 7 is 21 you have a remainder of 4 so let's see so based on that so let me rewrite this so 25 divided by 7 is equal to 3 remainder 4 so it's equal to 3 remainder 4 so based on this if we were to write out the grid we should have three rows of seven before we get to the 25 and then 25 should sit into the in the fourth column so if it's sitting in the fourth column it should be a Thursday so day 35 day 25 based on this little math we just did should be a Thursday let's see if that actually works out so let's go to 17 18 19 20 21 22 23 24 and 25 it is indeed a Thursday but it makes complete sense you could get seven rows in it before the row that gets to 25 and then in that row it's going to be the fourth entry because you have a remainder four one two three four it's going to be a Thursday so now we're ready to answer the question what is day three 300 going to be so let's just divide 300 let's just divide 300 by 7 and see where we get 7 goes into 30 4 times 4 times 4 times 7 is 28 subtract you get to bring down a 0 7 goes into 20 2 times 2 times 7 is 14 and then we get our remainder and now we care much more about the remainder 20 20 minus 14 is 6 so our remainder is 6 so if we think about what day of the week it is in its row it's going to be the 6th entry it's going to be the 6th column there's going to be 42 rows above it but we care about which entry it is in its row so day 300 is going to be the 6th day of the week the way we've written it out it is going to be it is going to be a Saturday