Factors and multiples: days of the week

Let's say that today-- and we'll call today Day 1-- Day 1 is a Monday. What I want to figure out is what is the Day 300 going to be? What day of the week will Day 300 be? And I encourage you to pause the video and think about that a little bit. So let's just write out the days of the week. You have Monday-- I'll do 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 this out. Monday is Day 1. Tuesday's Day 2. Wednesday, Day 3. 4, 5, 6, 7. I'll keep going. Day 8, well, that's going to be a Monday again. 9 10. I'm almost writing a calendar out. 11, 12, 13, 14, 15, 16. So this is kind of useful. I could just write it out if I wanted to figure out something Day 16 or Day 20. I could just write that out. But this isn't that helpful if I want to figure out Day 300 or especially wouldn't be helpful if I wanted to figure out Day 3,000. So can I come up with some mathematical way of thinking about what Day 300 is going to be? Well, as you see when I started drawing this grid here, this grid has rows. And 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, then 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. 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. What is the remainder when you divide 16 by 7? 16 divided by 7 is going to be 2. 2 times 7 is 14. You're going to have a remainder of 2. So when we divide, we've historically cared more about this 2. 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 first two just tells you 7 goes into 16 2 times. That's how many rows you have before getting to the 16. But then the remainder tells you in that row where is the 16? So the 16 is remainder 2. So the 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 other examples. Let's imagine Day 25. So let's just divide 25 by 7. So I'll do it right over-- I could do it. I'm going to 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 based on this, if we were to write out the grid, we should have three rows of 7 before we get to the 25. And then 25 should sit in the fourth column. So if it's sitting in the fourth column, it should be a Thursday. So 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 of 4. 1, 2, 3, 4. It's going to be a Thursday. So now we're ready to answer the question. What is Day 300 going to be? So let's just divide 300 by 7 and see where we get. 7 goes into 30 4 times. 4 times 7 is 28. Subtract, you get 2. 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 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 sixth entry. It's going to be the sixth 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 sixth day of the week, the way we've written it out. It is going to be a Saturday.