1 00:00:01,080 --> 00:00:04,835 Someone walks up to you on the street and says two thousand, nine hundred and forty-three. 2 00:00:04,835 --> 00:00:07,490 Quick! Is this divisible by nine? It's a matter of life and death! 3 00:00:07,490 --> 00:00:10,575 And you could say "well I can do this fairly quickly 4 00:00:10,575 --> 00:00:13,993 to figure out whether this is divisible by nine, I just have to add up the digits 5 00:00:13,993 --> 00:00:17,278 and figure out if the sum of the digits is multiple 6 00:00:17,278 --> 00:00:21,081 of nine or whether it is divisible by nine." 7 00:00:21,081 --> 00:00:26,549 Let's do that. Two plus nine plus four plus three. 8 00:00:26,549 --> 00:00:35,157 Two plus nine is eleven. Eleven plus four is fifteen. Fifteen plus three is eighteen. And eighteen is definitely divisible by nine. 9 00:00:35,157 --> 00:00:37,907 So this is going to be divisible by nine. 10 00:00:37,907 --> 00:00:42,245 And if you're unsure whether eighteen is divisible by nine, you can apply the same rule again. 11 00:00:42,245 --> 00:00:48,547 One plus eight is equal to nine. So that's definitely (repeated) by nine. 12 00:00:48,547 --> 00:00:53,996 And so the person can go save their life or whoever's life they are trying to save with that information. 13 00:00:53,996 --> 00:01:01,161 But this might lead you thinking how nice and useful this is. Why did that work? Does this work for all numbers or for only nine? 14 00:01:01,161 --> 00:01:07,879 I don't think this works for eight, or seven or eleven or seventeen. Why does this work for nine? 15 00:01:07,879 --> 00:01:10,947 This actually also works for three but we'll think about that in a future video. 16 00:01:10,947 --> 00:01:15,497 To realize that, we just have to rewrite two thousand, nine hundred and forty-three. 17 00:01:15,497 --> 00:01:25,946 So the two in 2943 in the thousands place so we can literally rewrite it as 2 x 1000. 18 00:01:25,946 --> 00:01:32,165 The nine is in the hundreds place so we can literally re-write it as 9 x 100. 19 00:01:32,165 --> 00:01:41,159 The four is in the tens place so literally the same thing as 4 x 10. And now finally we have our three in the ones place. 20 00:01:41,159 --> 00:01:44,497 We can write it as 3 x 1 or just three. 21 00:01:44,497 --> 00:01:48,347 So this says two thousand, nine hundred, forty and three. 22 00:01:48,347 --> 00:01:59,082 Now we can re-write each of these things as thousands, hundreds, tens as the sum of one plus something that is divisible by nine. 23 00:01:59,082 --> 00:02:04,554 So, a thousand, I can re-write as one plus nine-hundred ninety nine. 24 00:02:04,554 --> 00:02:10,917 I can re-write a hundred as one plus ninety-nine. 25 00:02:10,917 --> 00:02:16,252 I can re-write ten as one plus nine. 26 00:02:16,252 --> 00:02:22,082 And so two times a thousand is the same thing as two times one plus nine-hundred ninety nine. 27 00:02:22,082 --> 00:02:26,223 Nine times a hundred is the same thing as nine times one plus ninety-nine. 28 00:02:26,223 --> 00:02:31,021 Four times ten is the same thing as four times one plus nine. 29 00:02:31,021 --> 00:02:33,751 And I have this plus three over here. 30 00:02:33,751 --> 00:02:44,014 But now I can distribute, I can say that, well, this over here is the same as two times one which is two plus two times nine-hundred and ninety-nine. 31 00:02:44,014 --> 00:02:53,547 This thing, right over here, is the same thing as nine times one. Just to be clear doing this, I'm distributing the two over the first parentheses, these first two terms. 32 00:02:53,547 --> 00:03:00,215 Then the nine, I'm going to distribute again. It's going to be nine times one plus nine times ninety-nine. 33 00:03:00,215 --> 00:03:15,947 And then, over here I'm going to distribute the four. Four times one so plus four and then four times nine so plus four times nine. 34 00:03:15,947 --> 00:03:20,480 And then finally, I have this positive three, this plus three right over here. 35 00:03:20,480 --> 00:03:24,619 Now I'm just going to re-arrange this addition. So I'm just going to take all the terms 36 00:03:24,619 --> 00:03:29,014 and multiplying by nine hundred ninety nine. And I'm going to do that in orange. 37 00:03:29,014 --> 00:03:34,818 So I'll do this term, this term and this term right over here. 38 00:03:34,818 --> 00:03:43,416 And so I have two times nine hundred and ninety-nine plus that nine times ninety-nine plus four times nine. 39 00:03:43,416 --> 00:03:59,280 So that are those three terms and then I have plus two plus nine plus four and plus three. 40 00:03:59,280 --> 00:04:04,080 This is just the sum of our digits. This is what we did up here. 41 00:04:04,080 --> 00:04:10,966 And you might see where all of this is going. This orange stuff here, is this divisible by nine? 42 00:04:10,966 --> 00:04:14,903 Well it sure would definitely! Nine nine nine that's divisible by nine. 43 00:04:14,903 --> 00:04:17,474 So anything this multiplying this by divisible of nine. 44 00:04:17,474 --> 00:04:21,160 This is divisible by nine. This is definitely divisible by nine. 45 00:04:21,160 --> 00:04:24,592 Ninety nine. Regardless of whatever is being multiplying by nine. 46 00:04:24,592 --> 00:04:29,640 Whatever is multiplying by nine. Whatever is multiplying by ninety nine is divisible by nine. 47 00:04:29,640 --> 00:04:31,453 Because ninety nine is divisible by nine. 48 00:04:31,453 --> 00:04:33,746 So this works out. And same thing over here. 49 00:04:33,746 --> 00:04:38,226 You are always going to be multiplying by a multiple of nine. 50 00:04:38,226 --> 00:04:43,965 So all of this business right over here is definitely going to be divisible by nine. 51 00:04:43,965 --> 00:04:50,084 and so all for whole thing, and I all did was rewrote 2943 like this right over here. 52 00:04:50,084 --> 00:04:55,069 In order for whole thing to be divisible by nine. This part definitely is divisible by nine. 53 00:04:55,069 --> 00:05:00,139 In order for the whole thing does the rest of the sum and it has to be divisible by nine as well. 54 00:05:00,139 --> 00:05:08,449 So in order this whole thing, all of this has to be divisible by nine.